graph the function. then determine critical values, inflection points, interval(s) over which the function…

Explanation:

Step1: Find the first - derivative

The function is $y = e^{7x}$. Using the chain - rule, if $y = e^{u}$ and $u = 7x$, then $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=7$. So, $y^\prime=\frac{d}{dx}(e^{7x})=7e^{7x}$.

Step2: Find critical values

Critical values occur where $y^\prime = 0$ or $y^\prime$ is undefined. Since $y^\prime=7e^{7x}$ is never undefined and $7e^{7x}\gt0$ for all real $x$ (because $e^{7x}\gt0$ for all $x\in R$), there are no critical values.

Step3: Find the second - derivative

Differentiate $y^\prime = 7e^{7x}$ with respect to $x$. Using the chain - rule again, $y^{\prime\prime}=\frac{d}{dx}(7e^{7x}) = 49e^{7x}$.

Step4: Find inflection points

Inflection points occur where $y^{\prime\prime}=0$ or $y^{\prime\prime}$ is undefined. Since $y^{\prime\prime}=49e^{7x}$ is never undefined and $49e^{7x}\gt0$ for all real $x$, there are no inflection points.

Step5: Determine where the function is increasing

A function $y = f(x)$ is increasing when $y^\prime\gt0$. Since $y^\prime=7e^{7x}\gt0$ for all $x\in(-\infty,\infty)$, the function is increasing on $(-\infty,\infty)$.

Answer:

Choose the correct sketch of the given function: (The function $y = e^{7x}$ is an exponential growth function, which looks like option A (assuming option A is an increasing exponential - shaped curve)). List the critical values of $f(x)$: B. There are no critical values. List the inflection points of $f(x)$: B. There are no inflection points. Over what interval(s) is the function increasing? A. $(-\infty,\infty)$