-2x^{4}-3x^{3}+5x+x-5

Question

find extrema

Answer 1

I will assume you meant to write: f(x) = -2x⁴ - 3x³ +5x² + x - 5

Since this question is listed under "Algebra 2", I will assume you are not expected to use Calculus in solving the problem.

To do this problem, you will want to use a graphing calculator or a graphing utility on the web such as desmos.com

If using a TI-84 calculator, enter the function as Y₁ as follows:

Y₁ = -2x⁴ - 3x³ +5x² + x - 5

Then graph the function.

Use the "calc" button (blue "2nd" button, then "trace")

Select "4:maximum"

Using the cursor buttons scroll left of the first maximum value (first big hump) and select "enter" then scroll to the right of that maximum and "select "enter" then scroll to the top of the maximum and select "enter" and you will get the following: x = -1.785776 and y = 5.9043108

This means that the first maximum occurs at x = -1.785776 and the value of it is 5.9043108.

Repeat the process for the other maximum and minimum.

If you are using desmos, it's even easier. Type in the function and use your mouse to select the maximum or minimum and it will give you the x-value and y-value of each.

Answer 2

You want to start by taking the derivative of the given expression:

-8x^3 - 9x^2 + 5 + 1 = -8x^3 - 9x^2 + 6

We will have a minimum/maximum where the derivative is equal to 0. So we can set this expression equal to 0 and solve for x.

-8x^3 - 9x^2 + 6 = 0

To solve for x, I will type the equation into my calculator.

x = 0.65

Now, we need to determine if this is. a minimum or maximum. We will plug this x value back into our original expression.

-2(0.65)^4 - 3(0.65)^3 +5(0.65) + 0.65 - 5 = -2.28

If we choose a different x-value, we can compare that y-value to (0.65, -2.28) to see if this is a min or max. Let's choose x=1 to make this easier.

-2(1)^4 - 3(1)^3 +5(1) + 1 - 5 = -4

Hence, the extrema we found was a maximum: (0.65, -2.28)