f(x)=e^{7x}\nover what interval(s) is the function increasing? select the correct choice below and fill in…

f(x)=e^{7x}\nover what interval(s) is the function increasing? select the correct choice below and fill in any answer boxes with\na. (-∞,∞)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never increasing.\nover what interval(s) is the function decreasing? select the correct choice below and fill in any answer boxes wi\na.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never decreasing.\nover what interval(s) is the function concave up? select the correct choice below and fill in any answer boxes\na.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never concave up.\nover what interval(s) is the function concave down? select the correct choice below and fill in any answer box\na.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never concave down.

f(x)=e^{7x}\nover what interval(s) is the function increasing? select the correct choice below and fill in any answer boxes with\na. (-∞,∞)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never increasing.\nover what interval(s) is the function decreasing? select the correct choice below and fill in any answer boxes wi\na.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never decreasing.\nover what interval(s) is the function concave up? select the correct choice below and fill in any answer boxes\na.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never concave up.\nover what interval(s) is the function concave down? select the correct choice below and fill in any answer box\na.\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function is never concave down.

Answer

Explanation:

Step1: Find the first - derivative

The derivative of $y = e^{7x}$ using the chain - rule. If $y = e^{u}$ and $u = 7x$, then $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=7$. So, $y^\prime=\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=7e^{7x}$. Since $e^{7x}>0$ for all real $x$, then $y^\prime = 7e^{7x}>0$ for all $x\in(-\infty,\infty)$. So the function is increasing on $(-\infty,\infty)$.

Step2: Determine the decreasing interval

Since $y^\prime>0$ for all $x\in(-\infty,\infty)$, the function is never decreasing.

Step3: Find the second - derivative

Differentiate $y^\prime = 7e^{7x}$ with respect to $x$. Using the chain - rule again, if $y^\prime=7e^{u}$ and $u = 7x$, then $y^{\prime\prime}=49e^{7x}$. Since $e^{7x}>0$ for all real $x$, then $y^{\prime\prime}=49e^{7x}>0$ for all $x\in(-\infty,\infty)$. So the function is concave up on $(-\infty,\infty)$ and never concave down.

Answer:

  1. Increasing interval: A. $(-\infty,\infty)$
  2. Decreasing interval: B. The function is never decreasing.
  3. Concave - up interval: A. $(-\infty,\infty)$
  4. Concave - down interval: B. The function is never concave down.