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Questions: Algebra BusinessCalculus
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Find the value of \(\displaystyle c\) for \(\displaystyle f(x)=- 7 x^{2} - 4 x - 2\) that satisfies the Mean Value Theorem on \(\displaystyle [-9, 6]\).
The slope of the secant line is \(\displaystyle m = \frac{f(6) -f(-9)}{6-(-9)}=17\). Setting the derivative equal to the slope of the secant line gives the equation \(\displaystyle 17 = - 14 x - 4\). The solution is \(\displaystyle c = - \frac{3}{2}\).
\begin{question}Find the value of $c$ for $f(x)=- 7 x^{2} - 4 x - 2$ that satisfies the Mean Value Theorem on $[-9, 6]$.
\soln{9cm}{The slope of the secant line is $m = \frac{f(6) -f(-9)}{6-(-9)}=17$. Setting the derivative equal to the slope of the secant line gives the equation $17 = - 14 x - 4$. 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)=- 7 x^{2} - 4 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%207%20x%5E%7B2%7D%20-%204%20x%20-%202%20" alt="LaTeX: \displaystyle f(x)=- 7 x^{2} - 4 x - 2 " data-equation-content=" \displaystyle f(x)=- 7 x^{2} - 4 x - 2 " /> that satisfies the Mean Value Theorem on <img class="equation_image" title=" \displaystyle [-9, 6] " src="/equation_images/%20%5Cdisplaystyle%20%5B-9%2C%206%5D%20" alt="LaTeX: \displaystyle [-9, 6] " data-equation-content=" \displaystyle [-9, 6] " /> . </p> </p><p> <p>The slope of the secant line is <img class="equation_image" title=" \displaystyle m = \frac{f(6) -f(-9)}{6-(-9)}=17 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7Bf%286%29%20-f%28-9%29%7D%7B6-%28-9%29%7D%3D17%20" alt="LaTeX: \displaystyle m = \frac{f(6) -f(-9)}{6-(-9)}=17 " data-equation-content=" \displaystyle m = \frac{f(6) -f(-9)}{6-(-9)}=17 " /> . Setting the derivative equal to the slope of the secant line gives the equation <img class="equation_image" title=" \displaystyle 17 = - 14 x - 4 " src="/equation_images/%20%5Cdisplaystyle%2017%20%3D%20-%2014%20x%20-%204%20" alt="LaTeX: \displaystyle 17 = - 14 x - 4 " data-equation-content=" \displaystyle 17 = - 14 x - 4 " /> . 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>