# Find the maximum and minimum values for the function #f# defined by #f(x) = 2sinx + cos2x# in the interval #[0, pi/2]#?

Minimum:

Maximum:

Now to find the critical number,

Therefore,

And

Hence,

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To find the maximum and minimum values of the function ( f(x) = 2\sin(x) + \cos(2x) ) in the interval ([0, \frac{\pi}{2}]), we need to first find the critical points of the function within this interval.

The critical points occur where the derivative of the function is either zero or undefined.

First, we find the derivative of ( f(x) ): [ f'(x) = 2\cos(x) - 2\sin(2x) ]

Next, we find the critical points by setting the derivative equal to zero and solving for ( x ): [ 2\cos(x) - 2\sin(2x) = 0 ]

Solving this equation yields the critical points ( x = \frac{\pi}{6} ) and ( x = \frac{\pi}{2} ).

Next, we evaluate the function ( f(x) ) at the critical points and at the endpoints of the interval ([0, \frac{\pi}{2}]) to find the maximum and minimum values.

[ f(0) = 2\sin(0) + \cos(0) = 0 + 1 = 1 ] [ f\left(\frac{\pi}{6}\right) = 2\sin\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{3}\right) = 2\left(\frac{1}{2}\right) + \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2} ] [ f\left(\frac{\pi}{2}\right) = 2\sin\left(\frac{\pi}{2}\right) + \cos\left(\frac{\pi}{2}\right) = 2(1) + 0 = 2 ]

The maximum value of the function occurs at ( x = \frac{\pi}{2} ), where ( f(x) = 2 ).

The minimum value of the function occurs at ( x = 0 ), where ( f(x) = 1 ).

Therefore, the maximum value of the function in the interval ([0, \frac{\pi}{2}]) is ( 2 ) and the minimum value is ( 1 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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