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Let the definite integral be defined by the formula For more accurate result for c ∈ (a, b), we can use that for
Q.
Let the definite integral be defined by the formula For more accurate result for c ∈ (a, b), we can use that for
Q. then f(x) is of maximum degree
[Using L’Hospital rule]
Let the definite integral be defined by the formula For more accurate result for c ∈ (a, b), we can use that for
Q. and c is a point such that a < c < b, and (c, f(c)) is the point lying on the curve for which F(c) is maximum, then f '(c) is equal to
∴ F (c) is max. at the point (c, f (c)) where F’ (c) = 0
(For 46). Given the implicit function y^{3} – 3y + x = 0 For x ∈ (–∞, –2) ∪ (2,∞) it is y = f (x) real valued differentiable function, and for x ∈ (–2, 2) it is y = g(x) real valued differentiable function.
PASSAGE  2
Consider the functions defined implicitly by the equation y^{3} – 3y + x = 0 on various intervals in the real line. If x ∈(∞,  2) ∪ (2,∞) , the equation implicitly defines a unique real valued differentiable function y = f (x). If x ∈(2, 2) , the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.
Q.
PASSAGE  2
Consider the functions defined implicitly by the equation y^{3} – 3y + x = 0 on various intervals in the real line. If x ∈(∞,  2) ∪ (2,∞) , the equation implicitly defines a unique real valued differentiable function y = f (x). If x ∈(2, 2) , the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.
Q. The area of the region bounded by the curve y = f (x), the xaxis, and the lines x = a and x = b, where ∞ < a <b <2 , is
PASSAGE  2
Consider the functions defined implicitly by the equation y^{3} – 3y + x = 0 on various intervals in the real line. If x ∈(∞,  2) ∪ (2,∞) , the equation implicitly defines a unique real valued differentiable function y = f (x). If x ∈(2, 2) , the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.
Q.
For y = g(x), we have y^{3} – 3y + x = 0
⇒ [g (x) ]3  3[ g (x)] + x=0 ...(1)
Putting x = –x, we get
⇒ [g (x)]3  3[ g (x)]  x = 0 ...(2)
Adding equations (1) and (2) we get
For g(0) = 0, we should have g(x) + g(–x) = 0
[∵ From other factor we get g(0) = ±√3]
⇒ g(x) is an odd function
PASSAGE  3
Consider the function f : ( ∞,∞) →(∞,∞) defined by
Q. Which of the following is true?
⇒ (x^{2} + ax + 1) 2 f '(x) = 2a(x^{2} 1)
⇒ (x^{2} + ax +1)^{2} f "(x) + 2(x^{2} + ax+1)
(2 x + a) f '(x)=4ax ...(1)
Putting x = –1 in equation (1), we get
(2  a^{2}) f ''(1) =4a …(2)
Putting x = 1 in equation (1), we get
(2 + a)^{2} f "(1)=4a ...(3)
Adding equations (2) and (3), we get
(2 + a)^{2} f "(1) + (2  a)^{2}f "(1)= 0
PASSAGE  3
Consider the function f : ( ∞,∞) →(∞,∞) defined by
Q. Which of the following is true?
f '(x) = 0 ⇒ x = –1, 1 are the critical points.
∴ x = – 1 is a point of local maximum
and x = 1 is a point of local minimum.
PASSAGE  3
Consider the function f : ( ∞,∞) →(∞,∞) defined by
Q. Which of the following is true?
Now g '(x) >0 for e^{2x}  1> 0 ⇒ x > 0
and g '(x) <0 for e^{2x}  1 < 0 ⇒ x<0
∴ g'(x) is negative on (–∞ , 0) and posit ive on (0, ∞)
PASSAGE  4
f (x) = 1 + 2x + 3x^{2} + 4x^{3}.
Let s be the sum of all distinct real roots of f (x) and let t = s.
Q. The real numbers lies in the interval
f ( x) = 4x^{3} + 3x^{2} + 2x+1
∵ f (x) is a cubic polynomial
∴ It has at least one real root.
Also f '(x) =12x^{2} + 6x+ 2 = 2(6x^{2} +3x+1)
∴ f (x) is strictly increasing function
⇒ There is only one real root of f (x) = 0
∴ Real root lies between and hence
PASSAGE  4
f (x) = 1 + 2x + 3x^{2} + 4x^{3}.
Let s be the sum of all distinct real roots of f (x) and let t = s.
Q. The area bounded by the curve y = f (x) and the lines x = 0, y = 0 and x = t, lies in the interval
y = f (x), x = 0,y= 0 and x = t bounds the area as shown in the figure
∴ Required area is given by
PASSAGE  4
f (x) = 1 + 2x + 3x^{2} + 4x^{3}.
Let s be the sum of all distinct real roots of f (x) and let t = s.
Q. The function f'(x) is
f '(x) = 2(6x^{2} + 3x+ 1)
f ''(x) = 6 4x+ 1) ⇒ Critical point x = – 14
PASSAGE  5
Given that for each dt exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).
Q. The value of
PASSAGE  5
Given that for each dt exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).
Q.
PASSAGE  6
be a thrice differentiable function. Suppose that F(1) = 0, F(3) = –4 and F(x) < 0 for
Q. The correct statement(s) is(are)
f(x) = xF(x) ⇒ f ' (x) = F(x) + xF '(x)
f(2) = 2F(2) < 0,
(Q∵F '(x) < 0 ⇒ F is decreasing on
F(3) = –4)
f '(x) = F(x) + x F '(x)
For the same reason given above and F '(x) < 0 given.
∴ f '(x) ≠ 0, x∈(1, 3)
PASSAGE  6
be a thrice differentiable function. Suppose that F(1) = 0, F(3) = –4 and F(x) < 0 for
Q.
...(i)
⇒ 9(f ' (3) – F(3)) – (f ' (1) – F(1)) = 4
⇒ 9f ' (3) – 9 × (–4) – f ' (1) + 0 = 4
⇒ 9f ' (3) – f ' (1) + 32 = 0
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