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Thread: Critical numbers of Trig functions

  1. #1
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    Critical numbers of Trig functions

    Find the critical numbers on 0 < x < 2

    $\displaystyle f(x) = 4x - 4tan(x)$
    $\displaystyle f'(x) = 4 - sec^2(x)$

    f'(x) is undefined at $\displaystyle \frac{\pi}{2}$ and $\displaystyle \pi + \frac{\pi}{2}$

    I figured the next answer was $\displaystyle \frac{\pi}{3}$ and $\displaystyle \pi + \frac{\pi}{3}$ because that makes the derivative 0, but $\displaystyle n\pi + \frac{\pi}{3}$ isn't in the domain of the original function.

    So how would I go about finding the 0's?
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  2. #2
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    Quote Originally Posted by Open that Hampster! View Post
    Find the critical numbers on 0 < x < 2

    $\displaystyle f(x) = 4x - 4tan(x)$
    $\displaystyle f'(x) = 4 - sec^2(x)$

    correction ... $\displaystyle \textcolor{red}{f'(x) = 4 - 4\sec^2{x}}$
    yes, the original function and, therefore, its derivative are undefined at $\displaystyle x = \frac{\pi}{2}$

    $\displaystyle 4(1-\sec^2{x}) = 0$

    $\displaystyle \sec^2{x} = \pm 1$

    only solution in the interval [0,2] is x = 0
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    Ugh, I was messing the the math tags and it must have disappeared.

    The upper bound is 2 pi, not 2.
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    Quote Originally Posted by Open that Hampster! View Post
    Ugh, I was messing the the math tags and it must have disappeared.

    The upper bound is 2 pi, not 2.
    ok, then ...

    the original function and its derivative are undefined at $\displaystyle x = \frac{\pi}{2}$ and $\displaystyle x = \frac{3\pi}{2}$

    $\displaystyle \sec^2{x} = \pm 1$ at $\displaystyle x = 0$ , $\displaystyle x = \pi$ , and $\displaystyle x = 2\pi$
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