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Questions: Algebra BusinessCalculus
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Find the value of \(\displaystyle c\) for \(\displaystyle f(x)=6 x^{2} + 6 x + 5\) that satisfies the Mean Value Theorem on \(\displaystyle [-5, 5]\).
The slope of the secant line is \(\displaystyle m = \frac{f(5) -f(-5)}{5-(-5)}=6\). Setting the derivative equal to the slope of the secant line gives the equation \(\displaystyle 6 = 12 x + 6\). The solution is \(\displaystyle c = 0\).
\begin{question}Find the value of $c$ for $f(x)=6 x^{2} + 6 x + 5$ that satisfies the Mean Value Theorem on $[-5, 5]$.
\soln{9cm}{The slope of the secant line is $m = \frac{f(5) -f(-5)}{5-(-5)}=6$. Setting the derivative equal to the slope of the secant line gives the equation $6 = 12 x + 6$. The solution is $c = 0$. }
\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 + 5 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D6%20x%5E%7B2%7D%20%2B%206%20x%20%2B%205%20" alt="LaTeX: \displaystyle f(x)=6 x^{2} + 6 x + 5 " data-equation-content=" \displaystyle f(x)=6 x^{2} + 6 x + 5 " /> that satisfies the Mean Value Theorem on <img class="equation_image" title=" \displaystyle [-5, 5] " src="/equation_images/%20%5Cdisplaystyle%20%5B-5%2C%205%5D%20" alt="LaTeX: \displaystyle [-5, 5] " data-equation-content=" \displaystyle [-5, 5] " /> . </p> </p><p> <p>The slope of the secant line is <img class="equation_image" title=" \displaystyle m = \frac{f(5) -f(-5)}{5-(-5)}=6 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7Bf%285%29%20-f%28-5%29%7D%7B5-%28-5%29%7D%3D6%20" alt="LaTeX: \displaystyle m = \frac{f(5) -f(-5)}{5-(-5)}=6 " data-equation-content=" \displaystyle m = \frac{f(5) -f(-5)}{5-(-5)}=6 " /> . Setting the derivative equal to the slope of the secant line gives the equation <img class="equation_image" title=" \displaystyle 6 = 12 x + 6 " src="/equation_images/%20%5Cdisplaystyle%206%20%3D%2012%20x%20%2B%206%20" alt="LaTeX: \displaystyle 6 = 12 x + 6 " data-equation-content=" \displaystyle 6 = 12 x + 6 " /> . The solution is <img class="equation_image" title=" \displaystyle c = 0 " src="/equation_images/%20%5Cdisplaystyle%20c%20%3D%200%20" alt="LaTeX: \displaystyle c = 0 " data-equation-content=" \displaystyle c = 0 " /> . </p> </p>