Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = 5 − 24x + 4x2, [2, 4]

Answer 1

Answer:

The number $c = 3$ satisfies the conclusion of Rolle's Theorem for $f(x) = 5-24\cdot x +4\cdot x^{2}$.

Step-by-step explanation:

According to the Rolle's Theorem, for all function continuous on $[a,b]$, there is a value $c$ ($a \leq c\leq b$) such that:

$f'(c) = \frac{f(b)-f(a)}{b-a}$ (Eq. 1)

Where:

$f'(c)$ - First derivative of the function evaluated at $x = c$, dimensionless.

$a$, $b$ - Lower and upper bounds, dimensionless.

$f(a)$, $f(b)$ - Function evaluated at lower and upper bounds, dimensionless.

Let $f(x) = 5-24\cdot x +4\cdot x^{2}$, then upper and lower values are, respectively:

Lower bound ($a = 2$)

$f(2) = 5-24\cdot (2) +4\cdot (2)^{2}$

$f(2) = -27$

Upper bound ($b = 4$)

$f(4) = 5-24\cdot (4) +4\cdot (4)^{2}$

$f(4) = -27$

From Rolle's Theorem, we find that first derivative evaluated at $c$ is:

$f'(c) = \frac{-27-(-27)}{4-2}$

$f'(c) = 0$

Then, we find the first derivative of the function, equalize $x$ to $c$ and solve the resulting expression:

$f'(x) = -24+8\cdot x$

$-24+8\cdot c = 0$

 $c = 3$

The number $c = 3$ satisfies the conclusion of Rolle's Theorem for $f(x) = 5-24\cdot x +4\cdot x^{2}$.