Question

Given the function g(x) = x^4 - 4x^2 + 10, use the second derivative test to find the x-coordinates of all local minima. If there are multiple values, give them separated by commas. If there are no local minima, enter Ø. Write an exact answer; do not round. A) Ø B) -1, 1 C) -2, 2 D) -√2, √2

Answer 1

The x-coordinates of the local minima of the function g(x) = x^4 - 4x^2 + 10 are -√(2/3) and √(2/3).

Step-by-step explanation:

Given the function g(x) = x^4 - 4x^2 + 10, we can find the local minima by using the second derivative test. First, let's find the first and second derivatives of g(x).

The first derivative is g'(x) = 4x^3 - 8x.

The second derivative is g''(x) = 12x^2 - 8.

To find the x-coordinates of the local minima, we need to solve g''(x) = 0.

12x^2 - 8 = 0

12x^2 = 8

x^2 = 8/12

x^2 = 2/3

x = ±√(2/3)

Therefore, the x-coordinates of the local minima are -√(2/3) and √(2/3).