Question

let f(x)=xe^{9x}.
a. find the values of x for which the slope of the curve y = f(x) is 0.
b. explain the meaning of your answer to part (a) in terms of the graph of f.
a. find the values of x for which the slope of the curve y = f(x) is 0.
(simplify your answer. use a comma to separate answers as needed.)
b. explain the meaning of your answer to part (a) in terms of the graph of f. choose the correct answer.
a. the line tangent to the graph of f at the point(s) from part (a) is horizontal.
b. the line tangent to the graph of f at the point(s) from part (a) is vertical.
c. the value of the graph of f at the point(s) from part (a) do(es) not exist.
d. the value of the graph of f at the point(s) from part (a) is 0.

Explanation:

Step1: Differentiate using product - rule

The product - rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = x$ and $v = e^{9x}$. Then $u^\prime=1$ and $v^\prime = 9e^{9x}$. So, $f^\prime(x)=1\times e^{9x}+x\times9e^{9x}=e^{9x}(1 + 9x)$.

Step2: Set the derivative equal to 0

We want to find $x$ such that $f^\prime(x)=0$. Since $e^{9x}\gt0$ for all real $x$, we set $1 + 9x = 0$. Solving for $x$ gives $x=-\frac{1}{9}$.

Answer:

a. $-\frac{1}{9}$
b. A. The line tangent to the graph of f at the point(s) from part (a) is horizontal.