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Questions: Algebra BusinessCalculus
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Find the value of \(\displaystyle c\) for \(\displaystyle f(x)=- 7 x^{2} - x - 6\) that satisfies the Mean Value Theorem on \(\displaystyle [-9, 2]\).
The slope of the secant line is \(\displaystyle m = \frac{f(2) -f(-9)}{2-(-9)}=48\). Setting the derivative equal to the slope of the secant line gives the equation \(\displaystyle 48 = - 14 x - 1\). The solution is \(\displaystyle c = - \frac{7}{2}\).
\begin{question}Find the value of $c$ for $f(x)=- 7 x^{2} - x - 6$ that satisfies the Mean Value Theorem on $[-9, 2]$.
\soln{9cm}{The slope of the secant line is $m = \frac{f(2) -f(-9)}{2-(-9)}=48$. Setting the derivative equal to the slope of the secant line gives the equation $48 = - 14 x - 1$. The solution is $c = - \frac{7}{2}$. }
\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)=- 7 x^{2} - x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%207%20x%5E%7B2%7D%20-%20x%20-%206%20" alt="LaTeX: \displaystyle f(x)=- 7 x^{2} - x - 6 " data-equation-content=" \displaystyle f(x)=- 7 x^{2} - x - 6 " /> that satisfies the Mean Value Theorem on <img class="equation_image" title=" \displaystyle [-9, 2] " src="/equation_images/%20%5Cdisplaystyle%20%5B-9%2C%202%5D%20" alt="LaTeX: \displaystyle [-9, 2] " data-equation-content=" \displaystyle [-9, 2] " /> . </p> </p><p> <p>The slope of the secant line is <img class="equation_image" title=" \displaystyle m = \frac{f(2) -f(-9)}{2-(-9)}=48 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7Bf%282%29%20-f%28-9%29%7D%7B2-%28-9%29%7D%3D48%20" alt="LaTeX: \displaystyle m = \frac{f(2) -f(-9)}{2-(-9)}=48 " data-equation-content=" \displaystyle m = \frac{f(2) -f(-9)}{2-(-9)}=48 " /> . Setting the derivative equal to the slope of the secant line gives the equation <img class="equation_image" title=" \displaystyle 48 = - 14 x - 1 " src="/equation_images/%20%5Cdisplaystyle%2048%20%3D%20-%2014%20x%20-%201%20" alt="LaTeX: \displaystyle 48 = - 14 x - 1 " data-equation-content=" \displaystyle 48 = - 14 x - 1 " /> . The solution is <img class="equation_image" title=" \displaystyle c = - \frac{7}{2} " src="/equation_images/%20%5Cdisplaystyle%20c%20%3D%20-%20%5Cfrac%7B7%7D%7B2%7D%20" alt="LaTeX: \displaystyle c = - \frac{7}{2} " data-equation-content=" \displaystyle c = - \frac{7}{2} " /> . </p> </p>