Math / Calculus

QuestionAt least one of the answers above is NOT correct. (1 point) Find the coordinates of all extrema of $f(t)=4 t^{3}-60 t^{2}$ with domain $[-5, \infty)$ .

Studdy Solution

STEP 1

1. The function $f(t) = 4t^3 - 60t^2$ is a polynomial, which is continuous and differentiable everywhere.
2. The domain of $f(t)$ is restricted to $[-5, \infty)$ .
3. Extrema occur where the derivative is zero or undefined, and also at the endpoints of the domain.

STEP 2

1. Find the derivative of $f(t)$ .
2. Solve for critical points by setting the derivative equal to zero.
3. Evaluate the function at the critical points and the endpoint $t = -5$ .
4. Determine the nature of each critical point (local minimum, local maximum, or neither).
5. Identify and list the coordinates of all extrema.

STEP 3

Find the derivative of $f(t) = 4t^3 - 60t^2$ .
$f'(t) = \frac{d}{dt}(4t^3 - 60t^2)$
Using the power rule, the derivative is:
$f'(t) = 12t^2 - 120t$

STEP 4

Solve for critical points by setting $f'(t) = 0$ .
$12t^2 - 120t = 0$
Factor out the greatest common factor:
$12t(t - 10) = 0$
Set each factor equal to zero:
$12t = 0 \quad \Rightarrow \quad t = 0$
$t - 10 = 0 \quad \Rightarrow \quad t = 10$
The critical points are $t = 0$ and $t = 10$ .

STEP 5

Evaluate the function at the critical points and the endpoint $t = -5$ .
Calculate $f(-5)$ :
$f(-5) = 4(-5)^3 - 60(-5)^2$ $f(-5) = 4(-125) - 60(25)$ $f(-5) = -500 - 1500$ $f(-5) = -2000$
Calculate $f(0)$ :
$f(0) = 4(0)^3 - 60(0)^2$ $f(0) = 0$
Calculate $f(10)$ :
$f(10) = 4(10)^3 - 60(10)^2$ $f(10) = 4(1000) - 60(100)$ $f(10) = 4000 - 6000$ $f(10) = -2000$

STEP 6

Determine the nature of each critical point by using the first derivative test or evaluating the sign changes of $f'(t)$ .
For $t = 0$ :
- Check intervals around $t = 0$ using $f'(t) = 12t(t - 10)$ . - For $t < 0$ , $f'(t)$ is positive. - For $t > 0$ , $f'(t)$ is negative.
Thus, $t = 0$ is a local maximum.
For $t = 10$ :
- Check intervals around $t = 10$ . - For $t < 10$ , $f'(t)$ is negative. - For $t > 10$ , $f'(t)$ is positive.
Thus, $t = 10$ is a local minimum.

STEP 7

Identify and list the coordinates of all extrema:
- Local maximum at $t = 0$ with coordinate $(0, 0)$ . - Local minimum at $t = 10$ with coordinate $(10, -2000)$ . - Endpoint at $t = -5$ with coordinate $(-5, -2000)$ .
The coordinates of all extrema are $(0, 0)$ and $(10, -2000)$ . The endpoint $(-5, -2000)$ is not an extremum but is part of the domain evaluation.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms