for the given function find critical points and infliction

Question

f(x) = 2x^4 − 8x^3 + 1

1. Find all critical points.
2. Find all inflection points.
3. Determine where all local maximum(s) and minimum(s) occur using either the First Derivative Test or the Second Derivative Test.

Answer 1

Critical points on f are defined as anywhere f^'(x) = 0 or is undefined. Since f a polynomial, f^' is defined everywhere, so we can simply differentiate f (using power rule) to get f^', set it = 0, and solve:

f^'(x) = 8x³ - 24x² = 0

8x²(x - 3) = 0

x = 0 or x = 3 are the only two critical points for f.

Thinking about a sign chart for f^' will allow us to determine which type these are. f^'(x) is an odd degree poly with a positive leading coefficient, so it starts in Q III and finishes in QI. Thus, f^' is negative for x < 0. At x = 0, it has a root of even multiplicity, so its graph has a tangent root there. So f^' returns to being negative for 0 < x < 3, before crossing the x-axis at x = 3 and remaining positive for 3 < x.

f(x) has a flat point of inflection at (0,1) and a local minimum at (3,-53).

You can and should use Desmos or another graphing utility to confirm this analysis.