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1.5 p
Let
f(x)=(x-3)^(-2). Find all values of
c in
(1,4) such that
f(4)-f(1)=f^(')(c)(4-1). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

c=

$1$ . $1.5 p$ $\newline$ Let $f(x)=(x-3)^{-2}$ . Find all values of $c$ in $(1,4)$ such that $f(4)-f(1)=f^{\prime}(c)(4-1)$ . (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) $\newline$ $c=$

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Find the roots of factored polynomials

Full solution

Q.

1

.

1.5 p

\newline

Let

f(x)=(x-3)^{-2}

. Find all values of

c

in

(1,4)

such that

f(4)-f(1)=f^{\prime}(c)(4-1)

. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

\newline

c=

Calculate $f(4)$ : Calculate $f(4)$ using the function $f(x)=(x-3)^{-2}$ : $f(4) = (4-3)^{-2} = 1^{-2} = 1$ .
Calculate $f(1)$ : Calculate $f(1)$ using the function $f(x)=(x-3)^{-2}$ : $f(1) = (1-3)^{-2} = (-2)^{-2} = \frac{1}{4}$ .
Subtract $f(1)$ from $f(4)$ : Subtract $f(1)$ from $f(4)$ : $f(4) - f(1) = 1 - \frac{1}{4} = \frac{3}{4}$ .
Find the derivative of $f(x)$ : Find the derivative of $f(x)$ , $f'(x)$ : $f'(x) = \frac{d}{dx}[(x-3)^{-2}] = -2(x-3)^{-3}(1) = -\frac{2}{(x-3)^3}$ .
Set up the equation: Set up the equation using the Mean Value Theorem: $f(4)-f(1) = f'(c)(4-1)$ , so $\frac{3}{4} = \frac{-2}{(c-3)^3} \times 3$ .
Solve for $c$ : Solve for $c$ : $\frac{3}{4} = \frac{-6}{(c-3)^3}$ , so $(c-3)^3 = -8$ , but this is impossible since $(c-3)^3$ must be positive for $c$ in $(1,4)$ .

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