Question

The minimum value of $4e^{2x}+9e^{-2x}$ is

• A. $11$
• B. $12$
• C. $10$
• D. $14$

Answer 1

The correct option is B

$12$

Explanation for the correct option:

Step 1: Find the critical points of the given function.

A function $f\left(x\right)=4e^{2x}+9e^{-2x}$ is given.

Differentiate the given function with respect to $x$.

$f\text{'}\left(x\right)=8e^{2x}-18e^{-2x}$

Put $f\text{'}\left(x\right)$ equal to zero to find the critical points.

$$\begin{gathered} 8e^{2x}-18e^{-2x}=0 \\ \Rightarrow \frac{8{\left(e^{2x}\right)}^2-18}{e^{2x}}=0 \\ \Rightarrow 8{\left(e^{2x}\right)}^2-18=0 \\ \Rightarrow 8{\left(e^{2x}\right)}^2=18 \\ \Rightarrow {\left(e^{2x}\right)}^2=\frac{18}{8} \\ \Rightarrow e^{4x}=\frac{9}{4} \end{gathered}$$

Take $\log$ on both sides.

$$\begin{gathered} 4x=\log \left(\frac{9}{4}\right) \\ \Rightarrow x=\frac{\log \left(\frac{9}{4}\right)}{4} \end{gathered}$$

So, the critical point is $\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{4}$.

Step 2: Find the minimum value of the given function.

Since, the critical point is $\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{4}$.

Evaluate $f\left(x\right)$ for $x=\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{4}$ as follows:

$$\begin{gathered} f\left(x\right)=4e^{2·\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{4}}+9e^{-2·\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{4}} \\ \Rightarrow f\left(x\right)=4e^{\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{2}}+9e^{-\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{2}} \\ \Rightarrow f\left(x\right)=4e^{\log \left(\sqrt{\frac{9}{4}}\right)}+9e^{\log \left(\sqrt{\frac{4}{9}}\right)} \\ \Rightarrow f\left(x\right)=4e^{\log \left(\frac{3}{2}\right)}+9e^{\log \left(\frac{2}{3}\right)} \\ \Rightarrow f\left(x\right)=4·\frac{3}{2}+9·\frac{2}{3} \\ \Rightarrow f\left(x\right)=6+6 \\ \Rightarrow f\left(x\right)=12 \end{gathered}$$

So, the value of $f\left(x\right)$ for $x=\frac{\log \left({\displaystyle \frac{9}{4}}\right)}{4}$ is $12$.

Therefore, the minimum value of the given function is $12$.

Hence, option $B$ is the correct answer.