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Math Help - Finding Extrema

  1. #1
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    Finding Extrema

    I need help with this problem

    find all extrema in the interval [0, 2pi] for y=x - cos x. (write as points and mark each point as absolute, relative, or both)

    the first thing i did was to find the derivative of y=x - cos x. i did this and got dy/dt=1 +sin x. I then set this equal to 0 to find the critical points (i know that i didn't need to check where the derivative is undefined because it is defined at all x values since there is no denominator). The critical point i got was 3pi/2.
    I then plugged in my two endpoints (0 & 2pi) and my critical point (3pi/2) into the original equation to get the y values at each point. I got (0, -1), (2pi, 5.2831), and (3pi,2,4.7123).

    I'm pretty sure my points are right, but I don't know what they mean. I'm pretty sure that 0 is a relative min but can i also call it an absolute min? Are 3pi/2 and 2pi relative maxes? and can i call 2pi an absolute max?
    Please Help
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  2. #2
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    Quote Originally Posted by rawkstar View Post
    I need help with this problem

    find all extrema in the interval [0, 2pi] for y=x - cos x. (write as points and mark each point as absolute, relative, or both)

    the first thing i did was to find the derivative of y=x - cos x. i did this and got dy/dt=1 +sin x. I then set this equal to 0 to find the critical points (i know that i didn't need to check where the derivative is undefined because it is defined at all x values since there is no denominator). The critical point i got was 3pi/2.
    I then plugged in my two endpoints (0 & 2pi) and my critical point (3pi/2) into the original equation to get the y values at each point. I got (0, -1), (2pi, 5.2831), and (3pi,2,4.7123).

    I'm pretty sure my points are right, but I don't know what they mean. I'm pretty sure that 0 is a relative min but can i also call it an absolute min? Are 3pi/2 and 2pi relative maxes? and can i call 2pi an absolute max?
    Please Help
    \frac{dy}{dx} = 1 + \sin{x} \ge 0 for all x in [0,2\pi]

    that tells you the original function is increasing through the whole interval.

    endpoint minimum at x = 0 and endpoint maximum at x = 2\pi

    the graph of y "flattens out" at x = \frac{3\pi}{2}, but it is not an extrema.
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  3. #3
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    ok, then so not all critical points are extrema?
    also, would 0 and 2 be absolute mins/max or just relative?
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  4. #4
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    Quote Originally Posted by rawkstar View Post
    ok, then so not all critical points are extrema? that is sometimes true.

    also, would 0 and 2 be absolute mins/max or just relative?

    review your definitions ... relative extrema only occur inside the interval between the endpoints. this function has no relative extrema.

    you have two endpoint extrema since the function is increasing throughout the given interval ...

    f(0) = -1 is the absolute minimum.

    f(2pi) = 2pi - 1 is the absolute maximum.
    ...
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