The maximum value of the function f(x) = 3 si | vedclass

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(A) Given the function $f(x) = 3 \sin^{12} x + 4 \cos^{16} x$. Since $0 \le \sin^2 x \le 1$ and $0 \le \cos^2 x \le 1$, the powers $\sin^{12} x$ and $

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$4$

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(A) Given the function $f(x) = 3 \sin^{12} x + 4 \cos^{16} x$. Since $0 \le \sin^2 x \le 1$ and $0 \le \cos^2 x \le 1$, the powers $\sin^{12} x$ and $\cos^{16} x$ also lie in the interval $[0, 1]$. To find the maximum value, we test the boundary conditions of the trigonometric functions: Case $1$: If $\sin^2 x = 1$ (i.e.,$\cos^2 x = 0$), then $f(x) = 3(1)^6 + 4(0)^8 = 3$. Case $2$: If $\cos^2 x = 1$ (i.e.,$\sin^2 x = 0$), then $f(x) = 3(0)^6 + 4(1)^8 = 4$. Comparing the values, the maximum value of $f(x)$ is $4$.

The maximum value of the function $f(x) = 3 \sin^{12} x + 4 \cos^{16} x$ is

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