Find the critical points of f(x) = x^4 - 8x^2 + 16. (2021)

Find the critical points of f(x) = x^4 - 8x^2 + 16. (2021)

Step 1: Write down the function f(x) = x^4 - 8x^2 + 16.
Step 2: Find the derivative of the function, f'(x). The derivative is f'(x) = 4x^3 - 16x.
Step 3: Set the derivative equal to zero to find critical points: 4x^3 - 16x = 0.
Step 4: Factor the equation: 4x(x^2 - 4) = 0.
Step 5: Solve for x by setting each factor to zero: 4x = 0 gives x = 0, and x^2 - 4 = 0 gives x = ±2.
Step 6: List the critical points found: x = 0, x = 2, and x = -2.
Step 7: To confirm, evaluate f(2) = 0, which is a critical point.

Finding Critical Points – This involves taking the derivative of a function, setting it to zero, and solving for x to find points where the function's slope is zero.
Analyzing Derivatives – Understanding how to compute and interpret the first derivative to identify critical points and potential local maxima or minima.