Find all numbers c that satisfy the conclusion of Rolle's Theorem for the following function.

Question

Find all numbers c that satisfy the conclusion of Rolle's Theorem for the following function.

If there are multiple values, separate them with commas; enter N if there are no such values.

f(x)=x^2−9x+2, [0,9]

Answer 1

f(x)=x^2−9x+2, [0,9]

when, x = 0

f(x) = x^2 -9x +2

f(0) = y = 0 - 0 + 2 = 2

when, x=9

f(5) = y = 81 - 81 +2 = 2

thus, we have 2 points (0, 2) ; (9, 2)

slope,m = {2-(2)} / {9-0} = 0

hence, we have to calculate all the points,x where 0<x<5 and slope=0

f '(x) = 2x - 9 = 0

or, f '(c) = 2c - 9 = 0

c = 9/2 ( 0<x<9)

hence, the there is only one solution c=9/2 which satisfies Rolle's theore9