Question 30

$$a, b, c, d$$ are real constants in a $$f(x) = ax^{2025} + bx^{2023} + cx^{2021} + dx^{2019}$$ and $$f(-4) = 18$$. Then the maximum value of $$|f(4)| + |2\cos x|$$ is

Correct Answer: 20

Solution

Every power appearing in $$f$$ is odd, so $$f$$ is an odd function and $$f(4) = -f(-4) = -18$$, giving $$|f(4)| = 18$$. Since $$|2\cos x|$$ can be at most 2, the maximum value of the sum is $$18 + 2 = 20$$.

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