The function $f ( x ) = x ^ { 2 } - 6 x + 13$ satisfies the hypotheses of Rolle's Theorem on the interval $[ 0,6 ]$. Find all values of $c$ that satisfy the conclusion of the theorem.
A) 2,4
B) 2
C) 3,4
D) 3

Question

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Rolle's Theorem states that if a function $f$ is continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in $(a, b)$ such that $f'(c) = 0$. For $f(x) = x^2 - 6x + 13$, we first check if $f(0) = f(6)$. Indeed, $f(0) = 0^2 - 6*0 + 13 = 13$ and $f(6) = 6^2 - 6*6 + 13 = 13$, so the function satisfies the hypotheses of Rolle's Theorem. To find $c$, we differentiate $f(x)$ to get $f'(x) = 2x - 6$, and set $f'(x) = 0$ to solve for $x$. This gives $2x - 6 = 0$, leading to $x = 3$. Therefore, $c = 3$ is the value that satisfies the conclusion of Rolle's Theorem.

The Function f(x)=x2−6x+13f ( x ) = x ^ { 2 } - 6 x + 13f(x)=x2−6x+13

The Function $f ( x ) = x ^ { 2 } - 6 x + 13$