Question

Find the intervals in which the function 'f^' given by

f(x)=sin3x+3π, x∈[0,3π/2],

is strictly increasing or strictly decreasing.

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Answer 1

Given : f(x)=sin3x+3π, x∈[0,3π/2]

To Find : intervals in which the function is strictly increasing or strictly decreasing.

Solution:

f(x)=sin3x+3π

f'(x) = 3Cos3x  

x ∈ [0,3π/2]

=> 3x  ∈ [0 , 9π/2]

Strictly increasing

f'(x) = 3Cos3x    > 0

3x  ∈ (0 ,  π/2) ∪ ( 3π/2 , 5π/2) ∪  ( 7π/2 , 9π/2)

x  ∈ (0 ,  π/6) ∪ ( π/2 , 5π/6) ∪  ( 7π/6 , 3π/2)

Strictly decreasing

f'(x) = 3Cos3x  <  0

3x  ∈  (  π/2 , 3π/2)  ∪  ( 5π/2 , 7π/2)

x  ∈    ( π/6 ,  π/2) ∪  ( 5π/6 , 7π/6)

strictly increasing  (0 ,  π/6) ∪ ( π/2 , 5π/6) ∪  ( 7π/6 , 3π/2)

Strictly decreasing  ( π/6 ,  π/2) ∪  ( 5π/6 , 7π/6)

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