Question

Given f(x) = x⁸ - x², find the critical numbers.

a) x = 0, ±1
b) x = 0, ±√2
c) x = 0, ±3
d) x = 0, ±√3

Answer 1

To find the critical numbers of the function f(x) = x⁸ - x², we need to find the values of x where the derivative is equal to zero. The critical numbers are x = 0 and x = ±√2.

Step-by-step explanation:

To find the critical numbers of the function f(x) = x⁸ - x², we need to find the values of x where the derivative is equal to zero. The critical numbers are the values of x where the derivative changes sign or is undefined.

First, we find the derivative of f(x) using the power rule which states that the derivative of xⁿ is n*xⁿ⁻¹. The derivative of f(x) is f'(x) = 8x⁷ - 2x.

Next, we set the derivative equal to zero and solve for x. 8x⁷ - 2x = 0. Factor out x to get x(8x⁶ - 2) = 0. Therefore, the critical numbers are x = 0 and x = ±√2.