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  4. Let f be a twice differentiable function on (1,6) . If f (2)=8, f'(2)=5, f prime( x ) ≥ 1 and f''(x) ≥ 4, for all x ∈(1,6), then :

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Q. Let $f$ be a twice differentiable function on $(1,6) .$ If $f (2)=8, f'(2)=5, f ^{\prime}( x ) \geq 1$ and $f''(x) \geq 4,$ for all $x \in(1,6),$ then :

JEE MainJEE Main 2020Continuity and Differentiability

Solution:

$f(2)=8, f'(2)=5, f'(x) \geq 1, f''(x) \geq 4, \forall x \in(1,6)$

$f''(x)=\frac{f'(5)-f'(2)}{5-2} \geq 4$

$ \Rightarrow f'(5) \geq 17 \dots$(1)

$f'(x)=\frac{f(5)-f(2)}{5-2} \geq 1$

$ \Rightarrow f(5) \geq 11 \dots$(2)

$\overline{f'(5)+f(5) \geq 28}$