Find all values of c in the open interval (a, b) such that f'(c)=(f(b)-f(a))/(b-a) f(x)=4sin(x), [0,π]

There is only one value of c that satisfies the requirements.

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Work Shown:

Let's compute the function value at the given endpoints of the interval.

Which means,

We want to find all values of c such that the derivative is 0.

This can be rephrased into wanting to solve since the derivative of sine is cosine

In other words, turns into

Solving that equation leads to:

Use a reference table or the unit circle to determine this.

This is the value of c that satisfies f ' (c) = 0, such that . No other value of c works.

If you graphed f(x) = 4sin(x), and only focused on the interval [0,pi], then you'll find that there's a horizontal tangent at the point (pi/2, 4). Note how the endpoints have the same y value so that's why the average rate of change over [0,pi] is 0.

Side note: this is an application of Rolle's Theorem

yes because there is exactly one y value for every x value

for function there can't be more than one value for every x value