Find the minimum value of the function f(x) = x^4 - 8x^2 + 16.

Find the minimum value of the function f(x) = x^4 - 8x^2 + 16.

Step 1: Write down the function we want to minimize: f(x) = x^4 - 8x^2 + 16.
Step 2: Find the derivative of the function, which tells us the slope: 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: This gives us x = 0, x = 2, and x = -2.
Step 6: Evaluate the function at these critical points to find the minimum value: f(0), f(2), and f(-2).
Step 7: Calculate f(2): f(2) = 2^4 - 8(2^2) + 16 = 16 - 32 + 16 = 0.
Step 8: Calculate f(0): f(0) = 0^4 - 8(0^2) + 16 = 16.
Step 9: Calculate f(-2): f(-2) = (-2)^4 - 8(-2)^2 + 16 = 16 - 32 + 16 = 0.
Step 10: Compare the values: f(0) = 16, f(2) = 0, f(-2) = 0. The minimum value is 0.

Finding Critical Points – The process of taking the derivative of a function and setting it to zero to find points where the function may have local minima or maxima.
Second Derivative Test – A method to determine whether a critical point is a local minimum, local maximum, or saddle point by evaluating the second derivative at that point.
Evaluating Function Values – Calculating the function's value at critical points to determine the minimum or maximum value of the function.