Verify that the hypotheses of the Mean Value Theorem are satisfied for each of the functions on the given intervals, and find the value of a number(s) "c" which Mean Value Theorem guarantees

Question

For the function: f(x) = 5 - x on [1, 9]

Answer 1

f(9) = -4 and f(1) = 4

the slope of the secant line is (4 - -4)/(1-9)

= 8/-8 = -1

which is not surprising since the slope of

the line is -1, as is the derivative

MVT says there exists c in [1,9] such

that f'(c)=-1 = -1 = slope of secant line

so there are infinitely many such c...

for ANY real number c in [1,9]

the derivative is -1