let (f(x)=x^{2}+10x + 9). a. find the values of (x) for which the slope of the curve (y = f(x)) is 0. b…

let (f(x)=x^{2}+10x + 9). a. find the values of (x) for which the slope of the curve (y = f(x)) is 0. b. find the values of (x) for which the slope of the curve (y = f(x)) is 18. a. the slope of the curve is 0 at (x=) (simplify your answer. use a comma to separate answers as needed.)

let (f(x)=x^{2}+10x + 9). a. find the values of (x) for which the slope of the curve (y = f(x)) is 0. b. find the values of (x) for which the slope of the curve (y = f(x)) is 18. a. the slope of the curve is 0 at (x=) (simplify your answer. use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Find the derivative of $f(x)$

The derivative of $f(x)=x^{2}+10x + 9$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $f^\prime(x)=2x+10$.

Step2: Set the derivative equal to 0 to solve for part a

We want to find $x$ when $f^\prime(x)=0$. So we set up the equation $2x + 10=0$. Subtract 10 from both sides: $2x=-10$. Divide both sides by 2: $x=-5$.

Answer:

-5