Find the critical points of the function f(x) = x^4 - 8x^2 + 16. (2019)
Practice Questions
Q1
Find the critical points of the function f(x) = x^4 - 8x^2 + 16. (2019)
(0, 16)
(2, 0)
(4, 0)
(1, 9)
Questions & Step-by-Step Solutions
Find the critical points of the function f(x) = x^4 - 8x^2 + 16. (2019)
Step 1: Write down the function f(x) = x^4 - 8x^2 + 16.
Step 2: Find the derivative of the function, which is 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: The critical points are x = 0, x = 2, and x = -2.
Step 7: To find the corresponding y-values, evaluate f(2) and f(-2): f(2) = 2^4 - 8(2^2) + 16 = 0 and f(-2) = (-2)^4 - 8(-2)^2 + 16 = 0.
Step 8: The critical points are (0, f(0)), (2, 0), and (-2, 0).
Critical Points – Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection.
Derivative Calculation – Finding the derivative of a function is essential for determining critical points.
Function Evaluation – Evaluating the original function at critical points helps confirm their nature and coordinates.
