Maximum of Minimum Function

Question

Let $f(x) = \min\{2x^2, 52 - 5x\}$, where $x$ is any positive real number. Then the maximum possible value of $f(x)$ is

Accepted Answer

32. The maximum of $\min\{g(x), h(x)\}$ occurs where $g(x) = h(x)$ if one function is increasing and the other is decreasing. $2x^2 = 52 - 5x \Rightarrow 2x^2 + 5x - 52 = 0$. $2x^2 + 13x - 8x - 52 = 0 \Rightarrow x(2x + 13) - 4(2x + 13) = 0$. Since $x > 0$, $x = 4$. Maximum value = $2(4)^2 = 32$.

Answer

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Maximum of Minimum Function

Explanation

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