Question

Use a sign chart for $f\text{'}\text{'}$to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f\left(x\right)=e^{3x}\left(1-e^x\right)$

Short answer

The inflection points are$x=\ln \frac{16}{9}$and concave up on$\left(-\infty ,\ln \frac{16}{9}\right)$and concave down on$\left(\ln \frac{16}{9},\infty \right).$

Step-by-step solution

Step 1. Given Information.

$f\left(x\right)=e^{3x}\left(1-e^x\right)$is The given function.

Step 2. Second derivative.

On differentiating,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\frac{d}{dx}{e}^{3x}\left(1-e^x\right) \\ =3e^{3x}-4e^{4x} \\ {f}^{\text{'}\text{'}}\left(x\right)=\frac{d}{dx}\left(3e^{3x}-4e^{4x}\right) \\ =9e^{3x}-16e^{4x} \end{gathered}$$

Step 3. Sign chart.

Now,

$f\text{'}\text{'}\left(x\right)=0atx=\ln \frac{16}{9}$

Therefore, the sign chart will be,

$$\begin{gathered} Inflectionpoint:x=\ln \frac{16}{9} \\ Concaveup:\left(-\infty ,\ln \frac{16}{9}\right) \\ Concavedown:\left(\ln \frac{16}{9},\infty \right) \end{gathered}$$

Step 4. Verification.

The graph of the function is,