calculus

find all extrema in the interval [0, 2pi] if y=sinx + cosx

and also

a certain item sells for $30. if the cost of producing this item is given by C=.05^3 + 100, find the mariginal profit when x=10.

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  1. To find the extrema, do the First Derivative Test.

    y = sinx + cos x
    y'= cosx - sinx
    The extrema always occur when the derivative is zero, so set it equal to zero.

    y' = cosx - sinx = 0
    = cosx = sinx
    cosx/cosx = sinx/cosx
    1 = tanx
    x = pi/4, 5pi/4

    Make a sign chart...
    y' increases on (0, pi/4) and (5pi/4, 2pi)
    y' decreases on (pi/4, 5pi/4)
    A max occurs at pi/4, a min occurs at 5pi/4.

    The extrema are...
    Relative maximum = √2 @ x = pi/4
    Relative minimum = -√2 @ x = 5pi/4

    I think you've made a mistake on your second problem. There is no variable in the C function.

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  2. C=.05x^3+100

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