Given f(x) = e^{sin x} + e^{cos x}. The globa | vedclass

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(C) Given the function $f(x) = e^{\sin x} + e^{\cos x}$.To find the critical points,we set the derivative $f'(x) = 0$:$f'(x) = e^{\sin x} \cdot \cos x

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exists at infinitely many points

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(C) Given the function $f(x) = e^{\sin x} + e^{\cos x}$.To find the critical points,we set the derivative $f'(x) = 0$:$f'(x) = e^{\sin x} \cdot \cos x + e^{\cos x} \cdot (-\sin x) = 0$$e^{\sin x} \cos x = e^{\cos x} \sin x$$\frac{e^{\sin x}}{e^{\cos x}} = \frac{\sin x}{\cos x}$$e^{\sin x - \cos x} = 	an x$Since $f(x)$ is a periodic function with period $2\pi$,the solutions for $x$ occur periodically.At $x = \frac{\pi}{4}$,$e^{\sin(\pi/4) - \cos(\pi/4)} = e^0 = 1$ and $	an(\pi/4) = 1$. Thus,$x = \frac{\pi}{4}$ is a critical point.Due to the periodicity of $\sin x$ and $\cos x$,the function $f(x)$ repeats its values every $2\pi$.Therefore,the global maximum occurs at $x = 2n\pi + \frac{\pi}{4}$ for all integers $n$.Since there are infinitely many such points,the global maximum exists at infinitely many points.

Given $f(x) = e^{\sin x} + e^{\cos x}$. The global maximum value of $f(x)$

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