Question

Find the Absolute Max and Min over the Interval f(x)=x⁴-8x²-1 , [-3,4]

a) Absolute Max at (-3, 32), Absolute Min at (4, -255)
b) Absolute Max at (4, 255), Absolute Min at (-3, -32)
c) Absolute Max at (-3, -32), Absolute Min at (4, 255)
d) Absolute Max at (4, -255), Absolute Min at (-3, 32)

Answer 1

To find the absolute maximum and minimum of a function over an interval, we need to find the critical points and endpoints of the interval. In this case, the absolute maximum occurs at (-3, 32) and the absolute minimum occurs at (4, -255).

Step-by-step explanation:

To find the absolute maximum and minimum of a function over an interval, we need to find the critical points and endpoints of the interval.

Step 1: Find the critical points

To find the critical points of the function, we need to find where the derivative is equal to zero or undefined. Taking the derivative of f(x), we get f'(x) = 4x³ - 16x.

Setting f'(x) = 0, we can solve for x to find the critical points. In this case, the critical points are x = -2 and x = 2.

Step 2: Evaluate the function at the critical points and endpoints

Substitute the critical points and the endpoints of the interval (-3 and 4) into the function f(x) = x⁴ - 8x² - 1 to find their corresponding y-values.

Step 3: Determine the absolute maximum and minimum

Compare the y-values of the critical points and endpoints. The highest y-value is the absolute maximum, and the lowest y-value is the absolute minimum.

In this case, the absolute maximum occurs at (-3, 32) and the absolute minimum occurs at (4, -255).