If f: R → R is a twice differentiable function such that f''(x) > 0 for all x ∈ R, and f (1/2 ) = 1/2, f(1) = 1, then

If f: R → R is a twice differentiable function such that… — JEE Main 2017 Mathematics PYQ

JEE Main 2017

f: R \to R

is a twice differentiable function such that

f''(x) > 0

for all

x \in R

, and

f\!\left(\dfrac{1}{2}\right) = \dfrac{1}{2}

f(1) = 1

, then

A.
$f'(1) \leq 0$
B.
$f'(1) > 1$
C.
$0 < f'(1) \leq \dfrac{1}{2}$
D.
$\dfrac{1}{2} < f'(1) \leq 1$

Show answer and solution

Correct option: B

Solution

Since f''(x) > 0, f is strictly convex, meaning f' is strictly increasing.

Apply the Mean Value Theorem on [1/2, 1]: there exists c ∈ (1/2, 1) such that f'(c) = (f(1) - f(1/2))/(1 - 1/2) = (1 - 1/2)/(1/2) = 1.

Since f' is strictly increasing and c < 1, we have f'(1) > f'(c) = 1, so f'(1) > 1.

[JEE Main 2017]

Practise Application of Derivatives the way it is asked

A timed set of real JEE Main Application of Derivatives questions, graded as the exam grades. No sign-up.

Start a Application of Derivatives test

More: all JEE Main Application of Derivatives questions · JEE Main 2017 · Mathematics chapter list

JEE Main 2017 Mathematics question on Application of Derivatives