Question

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

• A. $f\left(5\right)+f\text{'}\left(5\right)\ge 28$
• B. $f\text{'}\left(5\right)+f"\left(5\right)\le 20$
• C. $f\left(5\right)\le 10$
• D. $f\left(5\right)+f\text{'}\left(5\right)\le 26$

Answer 1

The correct option is A

$f\left(5\right)+f\text{'}\left(5\right)\ge 28$

Explanation for correct option(s)

Option A: $f\left(5\right)+f\text{'}\left(5\right)\ge 28$

Given data:

$f\left(2\right)=8$

$f\text{'}\left(2\right)=5$

$f\text{'}\left(x\right)\ge 1$

$f"\left(x\right)\ge 4$

$x\in \left(1,6\right)$

Consider the given equation as,

$f\text{'}\left(x\right)\ge 1$

Integrate above with respect $x$ with limit $2$ to $5$ because $x\in \left(1,6\right)$, $\left(1,6\right)$ can not be differentiable.

$$\begin{gathered} {\int }_2^{5}f\text{'}\left(x\right)dx\ge {\int }_2^{5}1dx \\ {\left[f\left(x\right)\right]}_2^5\ge {\left[x\right]}_2^5 \\ f\left(5\right)-f\left(2\right)\ge 5-2 \\ f\left(5\right)-f\left(2\right)\ge 3 \end{gathered}$$

From the given data $f\left(2\right)=8$ then the above Equation becomes

$$\begin{gathered} f\left(5\right)-8\ge 3 \\ f\left(5\right)\ge 8+3 \\ f\left(x\right)\ge 11......\left(1\right) \end{gathered}$$

Then, consider the given equation as,

$f"\left(x\right)\ge 4$

Integrate above with respect $x$ with limit $2$ to $5$ because $x\in \left(1,6\right)$, $\left(1,6\right)$ can not be differentiable.

$$\begin{gathered} {\int }_2^{5}f"\left(x\right)dx\ge {\int }_2^{5}4dx \\ {\left[f"\left(x\right)\right]}_2^5\ge 4{\left[x\right]}_2^5 \\ f\text{'}\left(5\right)-f\text{'}\left(2\right)\ge 4\left(5-2\right) \end{gathered}$$

From the given data $f\text{'}\left(2\right)=5$ then the above Equation becomes

$$\begin{gathered} f\text{'}\left(5\right)-5\ge 4\left(5-2\right) \\ f\text{'}\left(5\right)-5\ge 4\times 3 \\ f\text{'}\left(5\right)-5\ge 12 \\ f\text{'}\left(5\right)\ge 5+12 \\ f\text{'}\left(x\right)\ge 17......\left(2\right) \end{gathered}$$

Adding Equation (1) and (2)

$$\begin{gathered} f\text{'}\left(x\right)+f"\left(x\right)\ge 11+17 \\ f\text{'}\left(x\right)+f"\left(x\right)\ge 28 \end{gathered}$$

Hence, the correct answer is Option A.