Question

Find all the x-values of the critical points and inflection points of the function f(x) =x^4 - 2ax^2 +b. Assume a and b are positive b. Find the values of the parameters a and b if f has a critical points.

Answer 1

The critical points and inflection points of the function f(x)=x^4 - 2ax^2 + b can be found by taking the first and second derivatives of the function. The critical points are x = 0, x = √a, and x = -√a. The inflection points are x = √(a/3) and x = -√(a/3).

Step-by-step explanation:

An inflection point occurs when the concavity of a graph changes. We can find the inflection points by taking the second derivative of the function and setting it equal to zero.

Given that f(x) = x^4 - 2ax^2 + b, the second derivative is f''(x) = 12x^2 - 4a. Setting this equal to zero, we get 12x^2 - 4a = 0. This gives us two critical points, x = √(a/3) and x = -√(a/3).

To find the x-values of the critical points, we need to take the first derivative of the function and set it equal to zero. The first derivative is f'(x) = 4x^3 - 4ax. Setting this equal to zero, we can factor out 4x to get 4x(x^2 - a) = 0. This gives us three critical points, x = 0, x = √a, and x = -√a.