Question

Find all values of c in the open interval (a, b) such that f'(c)=(f(b)-f(a))/(b-a)

f(x)=√(4-x), [-5,4]​

Answer 1

Answer: c = 7/4

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Work Shown:

Compute the function value at the endpoints

`f(x) = √(4-x)\\\\f(-5) = √(4-(-5)) = 3\\\\f(4) = √(4-4) = 0\\\\`

With a = -5 and b = 4, we have

`f'(c) = (f(b)-f(a))/(b-a)\\\\f'(c) = (f(4)-f(-5))/(4-(-5))\\\\f'(c) = (0-3)/(9)\\\\f'(c) = -(1)/(3)\\\\`

So,

`f(x) = √(4-x)\\\\f'(x) = -(1)/(2√(4-x))\\\\f'(c) = -(1)/(3)\\\\-(1)/(2√(4-c)) = -(1)/(3)\\\\`

Use algebra to solve for c

`-(1)/(2√(4-c)) = -(1)/(3)\\\\(1)/(2√(4-c)) = (1)/(3)\\\\3 = 2√(4-c)\\\\2√(4-c) = 3\\\\√(4-c) = (3)/(2)\\\\4-c = (9)/(4)\\\\c = 4-(9)/(4)\\\\c = (16-9)/(4)\\\\c = (7)/(4)\\\\`