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Questions: Algebra BusinessCalculus
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Find the value of \(\displaystyle c\) for \(\displaystyle f(x)=- 6 x^{2} + 6 x - 1\) that satisfies the Mean Value Theorem on \(\displaystyle [-8, 5]\).
The slope of the secant line is \(\displaystyle m = \frac{f(5) -f(-8)}{5-(-8)}=24\). Setting the derivative equal to the slope of the secant line gives the equation \(\displaystyle 24 = 6 - 12 x\). The solution is \(\displaystyle c = - \frac{3}{2}\).
\begin{question}Find the value of $c$ for $f(x)=- 6 x^{2} + 6 x - 1$ that satisfies the Mean Value Theorem on $[-8, 5]$.
\soln{9cm}{The slope of the secant line is $m = \frac{f(5) -f(-8)}{5-(-8)}=24$. Setting the derivative equal to the slope of the secant line gives the equation $24 = 6 - 12 x$. The solution is $c = - \frac{3}{2}$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the value of <img class="equation_image" title=" \displaystyle c " src="/equation_images/%20%5Cdisplaystyle%20c%20" alt="LaTeX: \displaystyle c " data-equation-content=" \displaystyle c " /> for <img class="equation_image" title=" \displaystyle f(x)=- 6 x^{2} + 6 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%206%20x%5E%7B2%7D%20%2B%206%20x%20-%201%20" alt="LaTeX: \displaystyle f(x)=- 6 x^{2} + 6 x - 1 " data-equation-content=" \displaystyle f(x)=- 6 x^{2} + 6 x - 1 " /> that satisfies the Mean Value Theorem on <img class="equation_image" title=" \displaystyle [-8, 5] " src="/equation_images/%20%5Cdisplaystyle%20%5B-8%2C%205%5D%20" alt="LaTeX: \displaystyle [-8, 5] " data-equation-content=" \displaystyle [-8, 5] " /> . </p> </p><p> <p>The slope of the secant line is <img class="equation_image" title=" \displaystyle m = \frac{f(5) -f(-8)}{5-(-8)}=24 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7Bf%285%29%20-f%28-8%29%7D%7B5-%28-8%29%7D%3D24%20" alt="LaTeX: \displaystyle m = \frac{f(5) -f(-8)}{5-(-8)}=24 " data-equation-content=" \displaystyle m = \frac{f(5) -f(-8)}{5-(-8)}=24 " /> . Setting the derivative equal to the slope of the secant line gives the equation <img class="equation_image" title=" \displaystyle 24 = 6 - 12 x " src="/equation_images/%20%5Cdisplaystyle%2024%20%3D%206%20-%2012%20x%20" alt="LaTeX: \displaystyle 24 = 6 - 12 x " data-equation-content=" \displaystyle 24 = 6 - 12 x " /> . The solution is <img class="equation_image" title=" \displaystyle c = - \frac{3}{2} " src="/equation_images/%20%5Cdisplaystyle%20c%20%3D%20-%20%5Cfrac%7B3%7D%7B2%7D%20" alt="LaTeX: \displaystyle c = - \frac{3}{2} " data-equation-content=" \displaystyle c = - \frac{3}{2} " /> . </p> </p>