March 25, 2017

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4. Given the function f defined by f(x) = cos2x for -π≤ x ≤π

a. Find the x-intercepts of the graph of f.
b. Find the x and y coordinates of all relative maximum points of f. Justify your answer.
c. Find the intervals on which the graph of f is increasing.

  • calculus - ,

    Make a rough sketch of y = cos 2x

    x-intercepts, when cos 2x = 0
    2x = π/2 or 2x = 3π/2
    x = π/4 or x = 3π/4
    period of cos2x is π, thus adding/subracting π to any answer will yield a new answer.
    π/4 - π = -3π/4
    3π/4 - π = -π/4

    x-intercepts are ±π/4, ±3π/4

    f'(x) = 2sin 2x
    = 0 for max/min
    2sin 2x = 0
    sin 2x= 0
    2x = 0 , π , 2π
    x = 0, π/2, π , again, since the period is π other answers are
    x = -π/2, 0, -π
    A quick look at your y = cos 2x will show you which of these are maximums and which are minimum
    Max: at x = -π, 0, π
    max points: (-π,2), (0,2), and (π,2)

    To show where y = cos 2x is increasing,
    f'(x) has to be positive
    f'(x) = -2sin 2x has to be positive
    so take a look where y = -2sin 2x lies above the x-axis
    Since you have the sketch of y = cos 2x, it is easy to see that the curve is increasing between -π/2 and 0 and again between π/2 and π

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