Find the x-coordinates of all critical points of the given function. Determine whether each critical point is a relative maximum, minimum, or neither by first applying the second derivative test, and, if the test fails, by some other method. f(x) = 5x^6 − 10x^4

Step-by-step explanation:

The first derivative is ...

  f'(x) = 30x^5 -40x^3 = 10x^3(3x^2 -4)

This will have zeros (critical points) at x=0 and x=±√(4/3).*

We don't need the second derivative to tell the nature of these critical points. Since the degree is even, the function is symmetrical about x=0. Since the leading coefficient is positive, it generally has a U-shape. This means the "outer" critical points will be minima, and the central one will be a local maximum.

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However, since we're asked to use the 2nd derivative test first, we find the 2nd derivative to be ...

  f''(x) = 150x^4 -120x^2 = 30x^2(5x^2 -4)

For x=0, f''(0) = 0 -- as we expect for a function with a high multiplicity of the root at that point. For x either side of zero, both the function and the second derivative are negative, indicating downward concavity. That is, x = 0 is a local maximum.

For x² = 4/3, the second derivative is positive, indicating upward concavity. At x = ±√(4/3), we have local minima.

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* The "simplified" equivalent to √(4/3) is (2/3)√3.