Science:Math Exam Resources/Courses/MATH110/April 2013/Question 03 (c)

MATH110 April 2013

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Question 03 (c)
Let $f(x)={\frac {\sin(x)}{e^{x}}}$ on the interval $[0,\pi ]$. In this question you will sketch this function. (c) Determine where ƒ is increasing and where it is decreasing. Does ƒ have any local extrema?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
To determine where $f(x)$ is increasing and decreasing, we should find the first derivative, its critical points, and do the first derivative test.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Using the quotient rule, we find: ${\begin{aligned}f'(x)&={\frac {(\cos x)(e^{x})-(\sin x)(e^{x})}{(e^{x})^{2}}}\\&={\frac {e^{x}(\cos x-\sin x)}{(e^{x})^{2}}}\\&={\frac {\cos x-\sin x}{e^{x}}}\end{aligned}}$ Setting $\displaystyle f'(x)=0$, we can clear the denominator and get: $\displaystyle \cos x-\sin x=0$ or, equivalently, $\displaystyle \cos x=\sin x$ This equation has infinitely many solutions, but only one which is in the interval [0,π]. The trig ratios $\cos x$ and $\sin x$ are equal on a 45-45-90 degree triangle, so the angle in the interval [0,π] that solves the above equation is 45 degrees, or in radian measure, $x=\pi /4$. Next, we check to see if there are any critical points where $f'(x)$ is undefined. The derivative would be undefined where the denominator is equal to zero, but because $e^{x}$ is always positive, we don't need to worry about this here. Thus we have one critical point at $x=\pi /4$. Performing the first derivative test, we get: $[0,\pi /4)$ $x=\pi /4$ $(\pi /4,\pi ]$ $\displaystyle f'(x)$ + 0 - $\displaystyle f(x)$ increasing max decreasing Hence f(x) has a maximum at $x=\pi /4$.

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