Question

Let $f$ be a function such that $f\left(1\right)=4$ and $f\text{'}\left(x\right)\ge 2$ for $1\le x\le 4$ , How small can $f\left(4\right)$ possible be?

• A. $8$
• B. $12$
• C. $16$
• D. $2$
• E. $10$

Answer 1

The correct option is E

$10$

Explanation for correct option(s)

Option E: $10$

Given data

$f\left(1\right)=4$

$f\text{'}\left(x\right)\ge 2$ for $1\le x\le 4$

Consider the given equation as,

$f\text{'}\left(x\right)\ge 2$ for $1\le x\le 4$

$$\begin{gathered} f\left(x\right)\ge 2x+c \\ f\left(1\right)\ge 2\times 1+c \\ f\left(1\right)\ge 2+c \end{gathered}$$

From the given data

$f\left(1\right)=4$

$$\begin{gathered} f\left(1\right)\ge 2+c \\ 4\ge 2+c \\ c=2 \end{gathered}$$

For $x=4$

$$\begin{gathered} f\left(x\right)\ge 2x+2 \\ f\left(4\right)\ge 2\times 4+2 \\ f\left(4\right)\ge 8+2 \\ f\left(4\right)\ge 10 \end{gathered}$$

Hence, The correct answer is Option E.