Mathematics

Mathematics

Problem

Let ff be the polynomial function f(x)=2x4−16x2+24.f(x)=2x^{4}-16x^{2}+24. Respond to the following prompts. (a) Calculate f′(x)f'(x). (b) Determine all critical points of ff. (c) Construct a sign chart for f′(x)f'(x) and state the intervals on which ff is increasing and the intervals on which it is decreasing. (d) Classify each critical point as the location of a local maximum, a local minimum, or neither. Provide justification based on your sign chart.
Intuitive explanation

The key to understanding this polynomial is recognizing it as a quadratic in disguise. Notice that f(x)f(x) contains only even powers of xx—you can rewrite it as f(u)=2u2−16u+24f(u) = 2u^2 - 16u + 24 where u=x2u = x^2. This substitution transforms a quartic into a much simpler quadratic, letting you find critical points, roots, and behavior using familiar techniques before translating back to the original variable.

Intuitive explanation

The key to understanding this polynomial is recognizing it as a quadratic in disguise. Notice that f(x)f(x) contains only even powers of xx—you can rewrite it as f(u)=2u2−16u+24f(u) = 2u^2 - 16u + 24 where u=x2u = x^2. This substitution transforms a quartic into a much simpler quadratic, letting you find critical points, roots, and behavior using familiar techniques before translating back to the original variable.

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