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Question

Find the intervals in which the function f given by f(x)=sin4x+cos4x, x[0,π/2] is
(a) is decreasing
(b) is increasing
Also, state whether x=π/4 is a point of local minima or local maxima ?

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Solution

f(x)=sin4x+cos4x, x[0,π/2]
f(x)=4sin3xcosx4cos3xsinx
=4sinxcosx(cos2xsin2x)
=2sin2xcos2x
=sin4x

Now, f(x)=0 gives x=0,π/4
The point x=π/4 divides the interval [0,π/2] into two disjoint intervals [0,π/4) and (π/4,π/2]

When 0x<π/4
04x<π
Since, siny is positive in 0 to π.
sin4x>0
sin4x<0
f(x)<0
f(x) is decreasing on [0,π/4)

When π/4<xπ/2
π<4x2π
Since, siny is negative in π to 2π.
sin4x<0
sin4x>0
f(x)>0
f(x) is increasing on (π/4,π/2]

x=π/4
for x<π/4, f(x)<0
for x>π/4, f(x)>0
x=π/4 is local minima.

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