To find the inflection points of the function f(x) = x^4 - 12x^2, we must first compute the second derivative of the function, which will indicate where the rate of change of the slope is zero.
1. Start by finding the second derivative of the function f(x) = x^4 - 12x^2. The second derivative is determined by differentiating the function twice with respect to x.
To get the first derivative, we apply the power rule, which states that the derivative of x^n is n*x^(n-1). Thus, f'(x) = 4x^3 - 24x.
Then, we differentiate again to find the second derivative: f''(x) = 12x^2 - 24, which simplifies to 12(x^2 - 2).
2. Next, we'll set this second derivative equal to 0 and solve for x to find the possible points of inflection. So, we'll solve the equation 12(x^2 - 2) = 0 for x.
Dividing through by 12 gives x^2 - 2 = 0.
Then, adding 2 to both sides gives x^2 = 2.
Finally, taking the square root of both sides yields x = sqrt(2) and x = -sqrt(2) as the possible inflection points.
3. However, these are only possible inflection points. We have to confirm them by showing that the function changes concavity at these points. This means that the second-derivative test may not conclusively determine whether these points are inflection points or not.
Therefore, we compute the third derivative of our function, f'''(x), to verify the inflection point. By taking the derivative of f''(x) = 12(x^2 - 2), we find that f'''(x) = 24x.
4. Now, we substitute each possible inflection point into our third derivative to ensure that the result is not equal to zero. First consider f'''(-sqrt(2)), which gives -24sqrt(2), and then consider f'''(sqrt(2)), which gives 24sqrt(2). Since neither of these is zero, we can confirm that x = sqrt(2) and x = -sqrt(2) are indeed inflection points of f.
So, the function f has inflection points at x = sqrt(2) and x = -sqrt(2).