Determine the local minima of f(x) = x^4 - 4x^2. (2021)

Determine the local minima of f(x) = x^4 - 4x^2. (2021)

Step 1: Write down the function we want to analyze: f(x) = x^4 - 4x^2.
Step 2: Find the derivative of the function, which tells us the slope: f'(x) = 4x^3 - 8x.
Step 3: Set the derivative equal to zero to find critical points: 4x^3 - 8x = 0.
Step 4: Factor the equation: 4x(x^2 - 2) = 0.
Step 5: Solve for x: This gives us x = 0 and x^2 - 2 = 0, which means x = ±2.
Step 6: Now we have three critical points: x = 0, x = 2, and x = -2.
Step 7: To find the local minima, we need to evaluate the function at these critical points: f(0), f(2), and f(-2).
Step 8: Calculate f(0): f(0) = 0^4 - 4(0^2) = 0.
Step 9: Calculate f(2): f(2) = 2^4 - 4(2^2) = 16 - 16 = 0.
Step 10: Calculate f(-2): f(-2) = (-2)^4 - 4(-2)^2 = 16 - 16 = 0.
Step 11: Compare the values: f(0) = 0, f(2) = 0, f(-2) = 0. All are equal, so we check the second derivative to confirm local minima.
Step 12: Find the second derivative: f''(x) = 12x^2 - 8.
Step 13: Evaluate the second derivative at the critical points: f''(0) = -8 (indicating a local maximum), f''(2) = 32 (indicating a local minimum), f''(-2) = 32 (indicating a local minimum).
Step 14: Conclude that the local minima occur at x = 2 and x = -2.

Critical Points – Identifying where the derivative of the function is zero to find potential local minima or maxima.
Second Derivative Test – Using the second derivative to determine the nature of the critical points (minima, maxima, or saddle points).