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Find the intervals on which $f(x) = 8\cos 4(x)$ decreases for $0 \le x \le π $.
What is the fast way to compute it?

FreeMind
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4 Answers4

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This should get you started:

  • Take the derivative of $f(x)$.
  • Find the intervals for which the derivative is negative.
John
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Compute the derivative $f'(x)$ of $f(x)$ and find the values of $x$ in the interval $0 \leq x \leq \pi$ for which $f'(x)$ is negative.

Null
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  • I know that, find the intervals please. – FreeMind Oct 03 '14 at 16:35
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    @FreeMind: First, you asked for the way to compute it, not what the intervals are. Second, that's not how we roll here. Show where you're stuck, and we can help, but we're not here to do it for you. – John Oct 03 '14 at 16:37
  • Looks like you need to brush up on your etiquette, Bub (@FreeMind). If you know how to find the zeroes of the derivative, it must be constantly positive or negative on each of the intervals between the zeroes. – MPW Oct 03 '14 at 16:44
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$x\mapsto \cos x$ is decreasing for $x\in[0,\pi]\cup[2\pi,3\pi]$, then $x\mapsto \cos 4x$ is decreasing for $x\in [0,\frac{\pi}{4}]\cup[\frac{\pi}{2},\frac{3\pi}{4}]$, therefore, $f$ is decreasing for $x\in [0,\frac{\pi}{4}]\cup[\frac{\pi}{2},\frac{3\pi}{4}]$

idm
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Hint: The quickest way is to sketch it by hand. All you need to do is perform some transformations on $y= \cos x$ and then simply read off the intervals where it's decreasing.

If you do it, you'll find that the answer is $0\leq x \leq \pi/4$ or $\pi/2 \leq x \leq 3\pi/4$.

Sheheryar Zaidi
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