Let $f$ be a function such that $f(1) = 4$ and $f^{'}(x) \geq 2$ for $1 \leq x

Question

Let $f$ be a function such that $f(1) = 4$ and $f^{'}(x) \geq 2$ for $1 \leq x < 4$. How small can $f(4)$ possibly be?

Dr. Abubacker · Accepted Answer

(d)
Given, $f^{'}(x) \geq 2$
On integrating both sides $w.r.t.x$, we get
$f(x) \geq 2x + c$
$f(1) \geq 2 	imes 1 + c$
$\Rightarrow c \leq 4 - 2 \Rightarrow c \leq 2$
$\Rightarrow f(x) \geq 2x + 2$
$	herefore\ f(4) \geq 2 	imes 4 + 2 = 10$

Q. Let $f$ be a function such that $f(1) = 4$ and $f^{'}(x) \geq 2$ for $1 \leq x < 4$. How small can $f(4)$ possibly be?

Explanation

Reviewed by Dr. Abubacker

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