Exams › JEE Advanced › Maths

Let f: R -> R be defined as f(x) = x + cos(x) + 2, and let g(x) be the inverse function of f(x). Find g'(3) + g''(3).

1. 1
2. 2
3. 3
4. 4

Correct answer: 1

Solution

f(0) = 0 + cos(0) + 2 = 0 + 1 + 2 = 3. So g(3) = 0. f'(x) = 1 - sin(x). g'(y) = 1/f'(g(y)). g'(3) = 1/f'(g(3)) = 1/f'(0) = 1/(1-sin(0)) = 1/(1-0) = 1. For g''(y): differentiate g'(y) = 1/f'(g(y)) w.r.t. y. g''(y) = -f''(g(y))*g'(y) / [f'(g(y))]². f''(x) = -cos(x). g''(3) = -f''(g(3))*(g'(3)) / [f'(g(3))]² = -f''(0)*1 / [f'(0)]² = -(-cos(0))*1/(1)² = -(-1)/1 = 1. Wait: f''(x) = -cos(x), f''(0) = -cos(0) = -1. g''(3) = -(-1)*1/(1)² = 1. g'(3) + g''(3) = 1 + 1 = 2.

Related JEE Advanced Maths questions

• The function f(x) = √|x² - 5| x + 6 + √8 + 2|x| - |x|² is defined as a real number for values of x within which range?
• The relation x² = xy can be described as:
• Consider the function f: R → R given by f(x) = (e^x - e^(-x)) / (e^x + e^(-x)). Which of the following statements is accurate?
• Consider the functions f(x) = max {1 + sin(x), 1 - cos(x)} for x in the interval [0, 2π], and g(x) = max {1, |x - 1|} for x belonging to the set of real numbers. Which of the following is true?
• Given f(x + f(y)) = f(x) + y, x, y ∈ R. Find f(1000).
• Find the range of fog(x) = f(g(x)) = f(4x(1 − x)).

⚔️ Practice JEE Advanced Maths free + battle 1v1 →