Find the critical values & determine the absolute maxima and minimum of the function on its respective intervals: g(x)=24x-6tanx on [0,1.5]

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You need to perform the first derivative test to tell what are the intervals where the function increases or decreases such that:

`f'(x) = 24 - 6/(cos^2 x)`

You need to solve the equation f'(x) = 0 such that:

`24 - 6/(cos^2 x) = 0 =gt 4 - 1/(cos^2 x) = 0`

`4cos^2 x - 1 = 0 =gt (2cos x - 1)(2cos x + 1) = 0`

`2 cos x - 1 = 0 =gt 2 cos x = 1 =gt cos x = 1/2`

`x = pi/3`

`2cos x + 1 = 0 =gt cos x = -1/2`

Since the cosine function is not negative in interval `[0,1.5], ` then there is no value for x such that `cos x = -1/2` .

Hence, the function increases over interval `[0,1.04]`  and it decreases over [`1.04,1.5` ], thus the function reaches its maximum at `x = pi/3 = 1.04` .

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