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pochemuha
2 years ago
14

Find the absolute extreme of the function on the closed interval. y=9e^xsinx , [0,pi]

Mathematics
1 answer:
Molodets [167]2 years ago
3 0

For convenience sake, I will let y=f(x)=9e^{x}\sin x

First, we evaluate the function at the endpoints of the interval.

f(0)=9e^{0}\sin(0)=0\\\\f(\pi)=9e^{\pi}\sin(\pi)=0

Then, we need to find the critical points.

We can start by taking the derivative using the power rule.

f'(x)=9e^{x} \frac{d}{dx} \sin x+9\sin x \frac{d}{dx} e^{x}=9e^{x}(\cos x+\sin x)

Setting this equal to 0,

9e^x (\cos x+\sin x)=0

Since 9e^x > 0, we can divide both sides by 9e^x.

\cos x+\sin x=0\\\\\sin x=-\cos x\\\\\tan x=-1\\\\x=\frac{3\pi}{4}

f\left(\frac{3\pi}{4} \right)=9e^{3\pi/4}\sin \left(\frac{3\pi}{4} \right)=\frac{9\sqrt{2}e^{3\pi/4}}{2}

So, the absolute minimum is \boxed{\left(\frac{3\pi}{4}, \frac{9\sqrt{2}e^{3\pi/4}}{\sqrt{2}} \right)} and the absolute minima are \boxed{(0,0), (\pi, 0)}

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