How do you determine all values of c that satisfy the mean value theorem on the interval [pi/2, 3pi/2] for # f(x) = sin(x/2)#?

Answer 1

The value is #c=pi#

#f(x) = sin(x/2)# is continuous on #[pi/2, 3pi/2]# and differentiable on #(pi/2, 3pi/2)#
Therefore there exists a #c# on #( pi/2, 3pi/2)# such that
#f'(c) = ( f(3pi/2) - f(pi/2) ) / ( 3pi/2 -pi/2)=>f'(c)=(f(3pi/2)-f(pi/2))/pi#
We know that #f(x) = sin(x/2)# hence #f(3pi/2) = sin (3pi/4) = sqrt2/2# #f(pi/2) = sin(pi/4) = sqrt2/2#

We notice that

#( f(3pi/2) - f(pi/2) ) / pi = (sqrt2/2 - sqrt2/2) / pi = 0#

But

#f'(x) = (1/2) cos(x/2)#
#f'(c) = (1/2) cos(c/2) = 0=>cos(c/2)=0=>c/2=pi/2=>c=pi#
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Answer 2

To determine all values of ( c ) that satisfy the Mean Value Theorem on the interval ( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] ) for ( f(x) = \sin\left(\frac{x}{2}\right) ), we first find the average rate of change of ( f(x) ) over the interval. Then, we find the derivative of ( f(x) ) and set it equal to the average rate of change to solve for ( c ).

  1. Average rate of change: [ \text{Average rate of change} = \frac{f\left(\frac{3\pi}{2}\right) - f\left(\frac{\pi}{2}\right)}{\frac{3\pi}{2} - \frac{\pi}{2}} ]

  2. Derivative of ( f(x) ): [ f'(x) = \frac{1}{2} \cos\left(\frac{x}{2}\right) ]

  3. Set ( f'(c) ) equal to the average rate of change: [ f'(c) = \frac{f\left(\frac{3\pi}{2}\right) - f\left(\frac{\pi}{2}\right)}{\frac{3\pi}{2} - \frac{\pi}{2}} ]

  4. Solve for ( c ).

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Answer from HIX Tutor

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.

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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