Question

If $f\left(x\right)=x^2+ax+5$ is increasing function in $\left(2,3\right)$ then the minimum value of $a$

• A. $2$
• B. $-4$
• C. $-2$
• D. $4$

Answer 1

The correct option is B

$-4$

Explanation for the correct option.

Find the minimum value of $a$.

For the function $f\left(x\right)=x^2+ax+5$, the derivative is given as: $f\text{'}\left(x\right)=2x+a$. Now as the function is increasing in $\left(2,3\right)$, so $f\left(x\right)>0$ and thus

$$\begin{gathered} 2x+a\ge 0 \\ \Rightarrow a\ge -2x \end{gathered}$$

The minimum value of $a$ in the interval $\left(2,3\right)$ will occur at $x=2$.

So the minimum value is $-2\times 2=-4$.

Thus, the minimum value of $a$ for which the function $f\left(x\right)=x^2+ax+5$ is increasing in the interval $\left(2,3\right)$ is $-4$.

Hence, the correct option is B.