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The minimum point of the function f(x) = (x2/3) – x is at 
  • a)
     x = 1 
  • b)
    x = -1
  • c)
     x = 0 
  • d)
    x = 1/√3
Correct answer is option 'A'. Can you explain this answer?

Avinash Sharma answered
Correct Answer :- a
Explanation : f(x) = (x^2/3) - x
f'(x) = 2/3(x-1/2) - 1
f"(x) = -1/3(x-3/2)
For critical points. f′(x)=0
=> 2/3(x-1/2) - 1 = 0 
f has minimum value of x = 1

  • a)
    ∞    
  • b)
    2    
  • c)
    0
  • d)
    1
Correct answer is option 'C'. Can you explain this answer?

Yash Patel answered
the squeeze theorem for this. Recall that sinx is only defined on −1≤sinx≤1. Therefore

Consider the following two statements about the function f(x) = |x|  
P: f(x) is continuous for all real values of x
 Q: f(x) is differentiable for all real values of x  
Which of the f oll owi ng is TRU E? 
  • a)
    P is true and Q is false  
  • b)
    P is false and Q is true  
  • c)
    Both P and Q are true  
  • d)
    Both P and Q are false  
Correct answer is option 'A'. Can you explain this answer?

Avinash Sharma answered
The graph of f(x) is  
f(x) is continuous for all real values of x   Lim |x| = Lim |x| = 0 
as can be seen from graph of |x|. 
and  Lim f(x) = +1 as can be seen from graph of |x| 
 x → 0+ 
Left deriva tive ≠ Rig ht derivative 
So |x| is continuous but not differentiable at x = 0. 

The distance between the origin and the point nearest to it on the surface z2 = 1 + xy is 
  • a)
    1
  • b)
    √3/2
  • c)
    √3
  • d)
    -2
Correct answer is option 'A'. Can you explain this answer?

Baishali Bajaj answered

or pr – q^2 = 4 – 1 = 3 > 0 and r = +ve

so f(xy) is minimum at (0,0)

Hence, minimum value of d^2 at (0,0)

d2 = x^2 + y^2 + xy + 1 = (0)^2 + (0)^2 + (0)(0) + 1 = 1

Then the nearest point is

z^2 = 1 + xy = 1+ (0)(0) = 1

or z = 1

If x is real, find the maximum value of (-x2 + 3x + 7)
  • a)
    36/5
  • b)
    37/7
  • c)
    37/4
  • d)
    36/7
Correct answer is option 'C'. Can you explain this answer?

Bijoy Mehra answered
Given Equation:
- x is real, find the maximum value of (-x^2 + 3x + 7)

Step 1: Find the vertex of the parabola
- The given equation is in the form of a quadratic equation, -x^2 + 3x + 7.
- To find the maximum value, we need to find the vertex of the parabola represented by this equation.
- The x-coordinate of the vertex is given by the formula: x = -b/2a, where a=-1 and b=3 in this case.
- Substituting the values of a and b, we get x = -3/(2*(-1)) = 3/2.
- Now, substitute x = 3/2 back into the equation to find the maximum value.

Step 2: Calculate the maximum value
- Substitute x = 3/2 into the equation: (-3/2)^2 + 3*(3/2) + 7
- Simplify the expression to find the maximum value: -9/4 + 9/2 + 7 = 37/4
Therefore, the maximum value of the given equation (-x^2 + 3x + 7) when x is real is 37/4. Hence, the correct answer is option 'C'.

The value of the directional derivative of the function θ (x, y, z) = xy2 + yz2 + zx2 at the point (2, -1, 1) in the direction of the vector p = i + 2j + 2k is
  • a)
    1
  • b)
    0.95
  • c)
    0.93
  • d)
    0.9
Correct answer is option 'A'. Can you explain this answer?

Vertex Academy answered
Given that,
ϕ = xy2 + yz2 + zx2
directional vector (p) = I + 2j + 2K
Directional derivative = 
∇ϕ at the point (2, -1, 1) is
∇ϕ = ((-1)2 + 2(2)(1)) î + (2(2)(-1) + (1)2) ĵ + (2(-1)(1) + (2)2)k̂
= 5î - 3ĵ + 2k̂
Directional derivative = 
= 5 - 6 + 4 / 3
= 1

The maximum value of f ( x) = (1 + cos x) sin x is
  • a)
    3
  • b)
    3√3
  • c)
    4
  • d)
    3√3/4
Correct answer is option 'D'. Can you explain this answer?

Radhika Sharma answered
The given function is:
f(x) = (1 - cos(x))sin(x)

To find the maximum value of the function:
We can find the maximum value of the function by finding the critical points and determining whether they are maximum or minimum points.

Finding the critical points:
The critical points occur when the derivative of the function is zero or undefined. Let's find the derivative of the given function.

f'(x) = (1 - cos(x))cos(x) + sin(x)(-sin(x))
= cos(x) - cos^2(x) - sin^2(x)
= cos(x) - (1 - sin^2(x))
= cos(x) - 1 + sin^2(x)
= sin^2(x) + cos(x) - 1

Simplifying the derivative:
To find the critical points, we need to solve the equation f'(x) = 0.

sin^2(x) + cos(x) - 1 = 0

Using the identity sin^2(x) = 1 - cos^2(x):
1 - cos^2(x) + cos(x) - 1 = 0

Simplifying further:
-cos^2(x) + cos(x) = 0

Factoring out cos(x):
cos(x)(-cos(x) + 1) = 0

Setting each factor to zero:
cos(x) = 0 or -cos(x) + 1 = 0

Solving the first equation:
cos(x) = 0
This occurs when x = π/2 or x = 3π/2.

Solving the second equation:
-cos(x) + 1 = 0
cos(x) = 1
This occurs when x = 0 or x = 2π.

Therefore, the critical points are x = π/2, 3π/2, 0, and 2π.

Determining the nature of critical points:
To determine whether the critical points are maximum or minimum points, we can use the second derivative test. Let's find the second derivative of the function.

f''(x) = d/dx (sin^2(x) + cos(x) - 1)
= 2sin(x)cos(x) - sin(x)

Using the identity 2sin(x)cos(x) = sin(2x):
f''(x) = sin(2x) - sin(x)

Simplifying the second derivative:
f''(x) = 2sin(x)cos(x) - sin(x)
= sin(x)(2cos(x) - 1)

Evaluating the second derivative at the critical points:
f''(π/2) = sin(π/2)(2cos(π/2) - 1)
= 1(2(0) - 1)
= -1

f''(3π/2) = sin(3π/2)(2cos(3π/2) - 1)
= -1(2(0) - 1)
= 1

f''(0

Suppose C is the closed curve defined as the circle x2 + y2 = 1 with C oriented anti-clockwise. The value of ∮(xy2dx + x2ydy) over the curve C equals ________
    Correct answer is between '-0.03,0.03'. Can you explain this answer?

    Engineers Adda answered
    Concept:
    Green’s theorem:
    Let R be a closed bounded region in the xy plane whose boundary C consists of finitely many
    smooth curves.
    Let F1(x, y) & F(x, y) be functions that are continuous and have continuous partial
    derivatives 
     
    ∂F1 / ∂y and ∂F2 / ∂x. Then
    Analysis:
    Given curve C: x2 + y2 = 1
    = 0

    Mathematically, the functions in Green’s theorem will be
    • a)
      Continuous derivatives
    • b)
      Discrete derivatives
    • c)
      Continuous partial derivatives
    • d)
      Discrete partial derivatives
    Correct answer is option 'C'. Can you explain this answer?

    Engineers Adda answered
    The Green’s theorem states that if L and M are functions of (x,y) in an open region containing D and having continuous partial derivatives then,
    ∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path taken anticlockwise.

    Using Green’s theorem, the value of the integral  , where C is the square, cut from the first quadrant by the lines x = 1 and y = 1, will be.
    • a)
      1
    • b)
      1/2
    • c)
      5/3
    • d)
      3/2
    Correct answer is option 'D'. Can you explain this answer?

    Sanya Agarwal answered
    Concept:
    If M(x,y), N(x,y), ∂N/∂y and ∂M/∂x be continuous functions over region R bounded by a simple closed curve c in x-y plane, then according to this theorem:
    It is used to simplify the vector integration.
    It gives the relation between the closed line and open surface integration.
    Calculation:
    Given:
    Comparing with the standard equation Mdx + Ndy; M = -y2 and N = xy.
    ∴ 
    = 3/2

    The following surface integral is to be evaluated over a sphere for the given steady vector field, F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j, and k as unit base vectors.
     , Where S is the sphere, x2 + y2 + z2 = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is
    • a)
      π
    • b)
      2π
    • c)
    • d)
      4π
    Correct answer is option 'A'. Can you explain this answer?

    Pioneer Academy answered
    Concept:
    Gauss divergence theorem:
    It states that the surface integral of the normal component of a vector function  taken over a closed surface ‘S’ is equal to the volume integral of the divergence of that vector function taken over a volume enclosed by the closed surface ‘S’.
    Calculation:
    Given:
    F = xi + yj + zk
    Stokes theorem:
    It states that the line integral of a vector field  around any closed surface C is equal to the surface integral of the normal component of the curl of vector  over an unclosed surface ‘S’.
    Green's theorem:
     

    The value of where ‘C’ is the curve passing through the point (0, √2) and satisfying the differential equation dy/dx = −2x/y is ________.
      Correct answer is between '-5,-4'. Can you explain this answer?

      Engineers Adda answered
      Concept:
      Green's theorem converts the line integral to a double integral. 
      Green's theorem transform the line integral in xy - plane to a surface integral on the same xy - plane
      If P and Q are functions of (x, y) defined in an open region then
      Application:
      Given the differential equation is:
      y dy = -2 xdx
      C = 1
      ∴ The equation of a curve is:
      This is an ellipse.
      On comparing with  we get:
      a = 1 and b = √2
      Now, the given integral is:
      (4y – 3x)dx + (3x + 2y) dy
      Here M = 4y – 3x and N = 3x + 2y
      Applying Greens theorem, we convert the line integral to a double integral, i.e.
      ∴ The given integral becomes:
      = 
      = - (Area under ellipse)
      = 
      = 

       With a 1 unit change in b, what is the change in x in the solution of the system of equations x + y = 2, 1.01 x + 0.99 y = b?  
      • a)
        Zero  
      • b)
        2 units  
      • c)
        50 units  
      • d)
        100 units 
      Correct answer is option 'C'. Can you explain this answer?

      Sahil Majumdar answered
       Given  x + y = 2               …………….. (i) 
      1.01 x + 0.99 y = b      …………….. (ii) 
      Multiply 0.99 is equation (i), and subtract from equation (ii), we get  

      The expression curl (grad f), where f is a scalar function, is  
      • a)
        Equal to ∇2f  
      • b)
        Equal to div (grad f)  
      • c)
         A scalar of zero magnitude  
      • d)
        A vector of zero magnitude  
      Correct answer is option 'D'. Can you explain this answer?

      Gaurav Kapoor answered
      The expression curl (grad f), where f is a scalar function, is a vector of zero magnitude.

      Explanation:

      1. Curl and Gradient
      To understand the expression curl (grad f), we need to first understand the concepts of curl and gradient.

      - Curl: The curl of a vector field measures the rotation or circulation of the field at a particular point. It is denoted by curl(F) or ∇ × F, where ∇ is the del operator.

      - Gradient: The gradient of a scalar function measures the rate of change or the slope of the function at a particular point. It is denoted by grad(f) or ∇f.

      2. Curl of Gradient
      Now, let's consider the expression curl (grad f), where f is a scalar function.

      - First, we find the gradient of the scalar function f, which is given by grad(f) = ∇f. The gradient is a vector field.

      - Next, we take the curl of the gradient, which is given by curl (grad f) or ∇ × (∇f).

      3. Curl of Gradient of a Scalar Function
      For any scalar function f, the curl of its gradient is always a vector of zero magnitude. This means that the curl (grad f) = 0.

      - The reason for this is that the curl measures the rotation or circulation of a vector field. However, the gradient of a scalar function is a conservative vector field, which means it has zero curl.

      - In other words, the gradient of a scalar function represents a field with no rotation or circulation. Therefore, taking the curl of such a field will always result in zero.

      4. Conclusion
      In conclusion, the expression curl (grad f), where f is a scalar function, is a vector of zero magnitude. This is because the gradient of a scalar function represents a field with no rotation or circulation, and taking the curl of such a field always results in zero.

      Find the greatest value of the directional derivatives of the function f = x2yz3 at (2, 1, -1) is
        Correct answer is between '13.0,13.7'. Can you explain this answer?

        Sanya Agarwal answered
        Concept:
        The greatest value of directional derivative = |∇ ϕ|
        Analysis:
        f = x2yz3  (2, 1, -1)
        ∇f = -4i - 4j + 12k
        |Δ f| = 13.26

        The directional derivative of ϕ = xy + yz + zx along the tangent to the curve at x = t, y = t2, z = t3 at P(1,1,1) is A/√14, then the value of A is:
          Correct answer is '12'. Can you explain this answer?

          Mahesh Yadav answered
          To find the directional derivative of the function f(x,y,z) = xy + yz + zx, we need to find the gradient vector of f at the point P(1,1,1) and then take the dot product of the gradient vector with the tangent vector of the curve at P.

          First, let's find the gradient vector of f:

          ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
          = (y+z, x+z, x+y)

          Now, let's find the tangent vector of the curve at P(1,1,1). We can do this by taking the derivative of each component of the curve with respect to t:

          r(t) = (x(t), y(t), z(t))
          = (t, t^2, t^3)

          dr/dt = (dx/dt, dy/dt, dz/dt)
          = (1, 2t, 3t^2)

          At P(1,1,1), t = 1, so the tangent vector at P is:

          dr/dt = (1, 2(1), 3(1)^2)
          = (1, 2, 3)

          Finally, we can find the directional derivative by taking the dot product of the gradient vector and the tangent vector:

          Directional derivative = ∇f · dr/dt
          = (y+z, x+z, x+y) · (1, 2, 3)
          = (1+1, 1+1, 1+2)
          = (2, 2, 3)

          Therefore, the directional derivative of f = xy + yz + zx along the tangent to the curve at P(1,1,1) is (2, 2, 3).

          The directional derivative of 1/r in the direction of is
          • a)
            1/r2
          • b)
            -1/r2
          • c)
            1/r3
          • d)
            -/r3
          Correct answer is option 'B'. Can you explain this answer?

          Pioneer Academy answered
          Concept:
          Let f(r) be a function then directional derivative of the function f(r) is given by: 
          Calculation:
          Given:
          f(r) = 1/r
          As we know that, if f(r) is a function then directional derivative of the function f(r)is given by: 
          f(r) = 1/r
          ∵ 
          Here, we have to find the directional derivative of f(r) in the direction of . It will be given by:

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