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The value of ‘a’ for which x^{3}  3x + a = 0 has two distinct roots in [0, 1] is given by
Let α, β ∈[0,1].f (x) is continuous on [a,b] & differentiable on (a,b) and f (α) = f (β) = 0
∴ c ∈ (α, β) such that f' (c) = 0 ⇒ c = ±1∉ (0,1)
The value of ‘c’ in Lagrange’s mean value theorem for f (x) = x (x 2)^{2} in [0, 1]
f '(c) = 0 2c(c  2) + (c  2)^{2} = 0
c = 2,2/3
∴ c = 2/3 (c ≠2)
For the function f (x) = x^{3}  6x^{2} + ax + b, if Roll’s theorem holds in [1, 3] with
f (1) = f (3) ⇒ a = 11
Find Value of ‘c’ by using Rolle’s theorem for f (x) = log (x^{2} + 2)  log 3 on [1,1]
The chord joining the points where x = p and x = q on the curve y = ax^{2} + bx + c is parallel to the tangent at the point on the curve whose abscissa is
Apply Lagrange’s theorm
The least value of k for which the function f(x) = x^{2} + kx + 1 is a increasing function in the interval 1 < x < 2
The interval in which f (x) = x^{3}  3x^{2}  9x + 20 is strictly decreasing
Given f (x) = x^{3}  3x^{2}  9x + 20
⇒ f '(x) = 3x^{2} 6x 9
⇒ f '(x) = 3(x  3)(x +1)
Thus, f (x) is strictly increasing for
x ∈ (∞,1) U (3, ∞) and strictly decreasing for x ∈ (1, 3)
f ' (x) = 0 ⇒ x = 1; f^{1} (x) does not exist at x = 2
∴ x = 1 and x = 2 are two critical points
The number of stationary points of f (x) = sin x in [0,2π] are
f (x) = sinx ⇒ f '(x) = cosx ⇒ f '(x) = 0
Therefore number of stationary points of f (x) in [0, 2π] is 2.
Local minimum values of the function
AM > GM
If the function has maximum at x =3, then the value of ‘a’ is
since f (x) has local maximum at x = 3 ⇒ f ' (3) = 0 and f ^{11} (3)< 0
The point at which f (x) = (x 1)^{4} assumes local maximum or local minimum value are
Therefore n = iv is even and f^{iv} (1) = 24> 0
Therefore f (x) has local minimum at x = 1.
The global maximum and global minimum of f (x) = 2x^{3}  9x^{2} + 12x + 6 in [0, 2]
Therefore global maximum M_{1} = max{f (0), f (1), f (2)}= 11
Global minimum
M_{2} = max{f (0), f (1), f (2)}= 6
If the percentage error in the surface area of sphere is k, then the percentage error in its volume is
If an error of is made in measuring the radius of a sphere then percentage error in its volume is
V% = 3(S%)
The height of a cylinder is equal to its radius. If an error of 1 % is made in its height. Then the percentage error in its volume is
h = r and v = ph^{3}; V% = 3( h%)
The slope of the normal to the curve given by
The line is a tangent to the curve then n ∈
Calculate slope
The points on the curve at which the tangent is perpendicular to xaxis are
dy/dx is not defined.
The point on the curve at which the tangent drawn is
The sum of the squares of the intercepts on the axes of the tangent at any point on the curve x ^{2/3} + y^{2/3}= a^{2/3} is
Equation of the tangent at p (θ) to
If the straight line x cos α + y sinα = p touches the curve at the point (a, b) on it, then
Find dy/dx and the equation of the tangent
If the curves x = y² and xy = k cut each other orthogonally then k² =
m_{1}.m_{2} =1
The angle between the curves y = x³ and
Find dy/dx to the two curves at (1, 1) they are m_{1} and m_{2}. Then
If the curves ay + x² = 7 and x³ = y cut orthogonally at (1, 1) then a =
Slope of the first curve at (1, 1) is Slope of the second curve at (1, 1) is m_{2} = 3
A particle moves along a line is given by then the distance travelled by the particle before it first comes to rest is
A particle is moving along a line such that s = 3t^{3}  8t + 1. Find the time ‘t’ when the distance ‘S’ travelled by the particle increases.
A particle moves along a line by S = t^{3}  9t^{2} + 24t the time when its velocity decreases.
130 videos359 docs306 tests

130 videos359 docs306 tests
