If f : R → R is a twice differentiable function such that f ′ ′ x > 0 for all x &#8712…

If f:R→R is a twice differentiable function such that f′′x>0 for all x∈R and f12=12, f1=1 then
  1. 0<f′1≤12
  2. f′1≤0
  3. f′1>1
  4. 12<f′1≤1

Solution

Using LMVT on  fx for x∈12, 1

f1-f121-12=f'c, where c∈12, 1

1-1212=f'c ⇒   f'c=1, where c∈12, 1

∵ f  ″ x > 0 ∀ x ∈ R
 

∴     f'x is an increasing function ∀ x∈R

∴ f  ′ 1 > 1

Asked in: JEE Advanced 2017 (Paper 2)

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