crealolobk

2021-12-31

Find the sationary points of the curve and their nature for the equation $y={e}^{x}\mathrm{cos}x$ for $0\le x\le \frac{\pi}{2}$

I derived it and got${e}^{x}(-\mathrm{sin}x+\mathrm{cos}x)=0$

$e}^{x$ has no solution but I don't know how to find the x such that $-\mathrm{sin}x+\mathrm{cos}x=0$

I derived it and got

boronganfh

Beginner2022-01-01Added 33 answers

Hint:

$-\mathrm{sin}x+\mathrm{cos}x=0$

If and only if

$\mathrm{tan}x=1$

If and only if

aquariump9

Beginner2022-01-02Added 40 answers

Since you are asking for $0\le x\le \frac{\pi}{2}$ it's quite trivial that $x=\frac{\pi}{4}$ is the only solution of sinx=cosx

Edit:

Reasoning in$0\le x\le \frac{\pi}{2}$ : notice that $\mathrm{cos}x=0$ only if $x=\frac{\pi}{2}$ and for $x=\frac{\pi}{2}$ we have $\mathrm{sin}\frac{\pi}{2}=1\ne 0=\mathrm{cos}\frac{\pi}{2}$

So we can say for sure that$x=\frac{\pi}{2}$ is not a solution of the equation $\mathrm{sin}x=\mathrm{cos}x$

Divide by$\mathrm{cos}\left(x\right)$ both sides of $\mathrm{sin}x=\mathrm{cos}x$ and you get $\mathrm{tan}x=1$ which is satiesfied in $0\le x\le \frac{\pi}{2}$ only for $x=\frac{\pi}{4}$

Edit:

Reasoning in

So we can say for sure that

Divide by

Vasquez

Expert2022-01-08Added 669 answers

For all x,

But we also have the identity

So your equation is also equivalent to

That is,

But

And the only value of x in

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