(a) Find the exact value of the minimum of f for

Find the exact value of the maximum of f for

(b) Find the exact value of x at which f increases most rapidly.

eozoischgc
2021-12-19
Answered

Consider the function below.

$f\left(x\right)={x}^{2}{e}^{-x}$

(a) Find the exact value of the minimum of f for$x\ge 0$ .

Find the exact value of the maximum of f for$x\ge 0$ .

(b) Find the exact value of x at which f increases most rapidly.

(a) Find the exact value of the minimum of f for

Find the exact value of the maximum of f for

(b) Find the exact value of x at which f increases most rapidly.

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Part a: Assume that the height of your cylinder is 8 inches. Consider A as a function of r, so we can write that as $A\left(r\right)=2\pi {r}^{2}+16\pi r$ . What is the domain of $A\left(r\right)$ ? In other words, for which values of r is $A\left(r\right)$ defined?

Part b: Find the inverse function to$A\left(r\right)$ . Your answer should look like $r=$ "some expression involving A".

$r\left(A\right)=$

Hints:

1) To calculate an inverse function, you need to solve for r.

2)Here you could start with$A=2\pi {r}^{2}+16\pi r$ . This equation is the same as $A=2\pi {r}^{2}+16\pi r-A=0$ . Do you recognize this as a quadratic equation $a{x}^{2}+bx+c=0$ where the variable x is r? The coefficients would be $2\pi$ for a, $16\pi$ for b, and $-A$ for c.

3)You can solve for r using the quadratic formula even though the constant term c is a symbol here.

Part c: If the surface area is 225 square inches, then what is the radius r? In other words, evaluate$r\left(225\right)$ . Round your answer to 2 decimal places.

Need Part C

Part b: Find the inverse function to

Hints:

1) To calculate an inverse function, you need to solve for r.

2)Here you could start with

3)You can solve for r using the quadratic formula even though the constant term c is a symbol here.

Part c: If the surface area is 225 square inches, then what is the radius r? In other words, evaluate

Need Part C

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