"Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $\ge$ 1 and f''(x) $\ge$ 4, for all x $\in$ (1, 6), then :"

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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),

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

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

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $\ge$ 1 and f''(x) $\ge$ 4, for all x $\in$ (1, 6), then :

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Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $\ge$ 1 and f''(x) $\ge$ 4, for all x $\in$ (1, 6), then :

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