Consider the function on the interval (0, 2pi). f(x) = sin x cos x + 4 Find the

Question

Consider the function on the interval (0, 2pi). f(x) = sin x cos x + 4 Find the open interval(s) on which the function is increasing or decreasing Increasing: (0, pi/4) (pi/4, 3pi/4) 3pi/4, 5pi/4) (5pi/4,7pi/4) (7pi/4, 2pi) Decreasing: (0, pi/4) (pi/4, 3pi/4) (3pi/4, 5pi/4) (5pi/4, 7pi/4) (7pi/4, 2pi) Apply the First Derivative Test to identify all relative extrema, relative maxima (x, y) = (smallest x-value) (x,y) = (largest x-value) relative minima (x, y) = (smallest x-value) (x,y) = (largest x-value)

Explanation / Answer

f(x)=sinxcosx+4=(1/2)sin2x

as sinx is incresing at (0,pi/2), (3pi/2,5pi/2),(7pi/2,4pi)

hence (1/2)sin2x+4 will be increasing in (0,pi/4) ,(3pi/4,5pi/4) and (7pi/4,2pi)

decreasing in other interval of (pi/4,3pi/4),(5pi/4,7pi/4_

differentiating f(x)

df(x)/dx=cos2x which is increasing and decreasing in opposite direction

hence relative maxima for smallest x=pi/4

    and for largest is at 5pi/4

also relative minima for smallest =3pi/4

and for largest =7pi/4

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