Final answer:
The function f(x) = 2x⁴ - 4x² - 6 has three critical points, which are found by setting the first derivative to zero. The critical points are at x = 0, x = 1, and x = -1.
Step-by-step explanation:
To find the number of critical points for the function f(x) = 2x⁴ - 4x² - 6, we must find the function's derivatives and determine where they are equal to zero or undefined. The critical points occur where the first derivative f'(x) is equal to zero or does not exist.
The first step is to take the derivative of f(x) which gives us f'(x) = 8x³ - 8x. To find the critical points, set f'(x) equal to zero: 8x³ - 8x = 0. Factoring out the common terms, we get 8x(x² - 1) = 0, and further factoring we get 8x(x - 1)(x + 1) = 0. This yields three critical points: x = 0, x = 1, and x = -1.
Since all three critical points come from setting the derivative to zero and there are no points where the derivative is undefined, f(x) has a total of three critical points.