Let f be a twice differentiable function on (1,6).

Question

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f'(2)=5$, $f'(x)\ge1$ and $f''(x)\ge4$, for all $x\in(1,6)$, then

Accepted Answer

Given, $$f'(x) \ge 1$$

$$ \therefore $$ $$\int_2^5 {f'(x)} dx\, \ge \,\int_2^5 {dx} $$

$$ \Rightarrow f(5) - f(2) \ge 3$$

$$ \Rightarrow f(5) - 8 \ge 3$$

$$ \Rightarrow f(5) \ge 11$$ ...(1)

Also, $$f''(x) \ge 4$$

$$ \therefore $$ $$\int_2^5 {f''(x)} dx\, \ge \,\int_2^5 {4dx} $$

$$ \Rightarrow f'(5) - f'(2) \ge 4(3)$$

$$ \Rightarrow f'(5) - 5 \ge 12$$

$$ \Rightarrow f'(5) \ge 17$$ ...(2)

From (1) and (2),

$$f'(5) + f'(5) \ge 11 + 17$$

$$ \Rightarrow f'(5) + f'(5) \ge 28$$

Solutions · 1