Exams › JEE Main › Maths
If f and g are differentiable functions in [0, 1] satisfying f(0) = 2 = g(1), g(0) = 0 and f(1) = 6, then for some c ∈ ]0, 1[:
- f'(c) = 2g'(c)
- 2f'(c) = g'(c)
- 2f'(c) = 3g'(c)
- f'(c) = g'(c)
Correct answer: f'(c) = 2g'(c)
Solution
The correct option is based on the application of the Mean Value Theorem, which implies that there exists a point c in the interval (0, 1) where the rates of change of the functions f and g are related by their respective changes over the interval. Given the values of f and g at the endpoints, the relationship f'(c) = 2g'(c) reflects the proportional change between the two functions.
Related JEE Main Maths questions
- Suppose f(x)=α(x)β(x)γ(x) for every real x, where α(x), β(x), and γ(x) are differentiable. If f'(2)=18f(2), α'(2)=3α(2), β'(2)=-4β(2), and γ'(2)=kγ(2), then what is the value of k?
- For the piecewise-defined function
f(x) = { [sin((a+1)x) + sin x]/x, x < 0; c, x = 0; [√(x + bx²) - √x]/(b x^(3/2)), x > 0 }
to be continuous at x = 0, which values of a, b, and c are required?
- Let f(x) and g(x) be continuously differentiable functions such that f'(x) = g(x) and f''(x) = -f(x). Define h(x) = [f(x)]² + [g(x)]². If h(0) = 5, what is the value of h(10)?
- For what value of p can the function
f(x) = { ((4^x - 1)³)/(sin x · [log|1 + x²/3|]^p), x ≠ 0; 12(log 4)³, x = 0 }
be made continuous at x = 0?
- Consider the function f: R -> R given by f(x) = max{x, x³}. At which points is f(x) not differentiable?
- Consider the function
f(x)=[(1-sin³ x)/(3cos² x),, x<(π)/(2),; [4pt]
p,, x=(π)/(2),; [4pt]
q(1-sin x),, x>(π)/(2).]
If f is continuous at x=(π)/(2), then the pair (p,q) is
⚔️ Practice JEE Main Maths free + battle 1v1 →