Verify rolles theorem and find all values of c

Question

Verify that rolle's theorem can be applied to the function f(x)=x^3-8x^2+17x-10 on the interval [1,5]. Then find all values of c in the interval such that f '(c)=0.

Answer 1

The criteria for Rolle's Theorem to be valid is:

1) The function be continuous on the interval. Polynomials are continuous for all real numbers so "check" for that one.

2) The derivative exists on the interval. The derivative is also a polynomial so it exists for all real numbers so "check" that one off.

3) f(1) must equal f(5). f(1) = 0 and f(5) = 0 so "check" that one off as well

Based on the three "checks" above, Rolle's Theorem can be applied.

To find the values of "c" on [1, 5] such that f '(c) = 0, we must take the derivative and set it equal to zero:

Using the power rule, f '(x) = 3x² - 16x + 17

Setting f ' equal to zero and solving (using the quadratic formula):

3x² - 16x + 17 = 0

x = [16 ± √((-16)² - 4•3•17)]/2•3

x = (16 ± √52)/6

x = (16 ± 2√13)/6

x = (8 ± √13)/3

The approx values are x = 1.465 and x = 3.869 and both ot these vales are on the interval [1, 5] so the values of "c" are c = (8 - √13)/3 and c = (8 + √13)/3