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Find monotonic rising interval of an exponential function

Question:

If function y = 2sin2x, find its monotonic rising interval.

Solution:

Because the function y = 2x is a monotonic rising function. So, the monotonic rising interval of y = 2sin2x is the monotonic rising interval of y = sin2x.

Now we are going to find the monotonic rising interval of the function y = sin2x. Look the graph of y = sin x, its monotonic rising interval is:

2pi k - pi/2 <= x <= pi/2 + 2pi k

So, the monotonic rising interval of the function y = sin2x is:

2pi k - pi/2 <= 2x <= pi/2 + 2pi k

pi k - pi/4 <= x < pi/4 + pi k

when k = 0, its monotonic rising interval is: -pi/4 <= x <= pi/4

when k = 1, its monotonic rising interval is: pi - pi/4 <= x <= pi/4 + pi, which is, 3pi/4 <= x <= 5pi/4

find monotonic rising interval of an exponential function

The graph above is y = 2sin2x. Its first monotonic rising interval is from x = -pi/4 to x = pi/4. It’s the second rising interval is from x = 3pi/4 to 5pi/4.

when x = -pi/4, y = 2sin2(-pi/4) = 2sin(-pi/2) = 2-sin(pi/2) = 2-1 = 1/2 = 0.5

when x = pi/4, y = 2sin2(pi/4) = 2sin(pi/2) = 21 = 2

when x = 3pi/4, y = 2sin2(3pi/4) = 2sin(3pi/2) = 2-1 = 1/2 = 0.5

when x = 5pi/4, y = 2sin2(5pi/4) = 2sin(5pi/2) = 2sin(2pi + pi/2) = 2sin(pi/2) = 21 = 2.

Watch the video for more details.