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Questions: Algebra BusinessCalculus
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Find the value of \(\displaystyle c\) for \(\displaystyle f(x)=- 2 x^{2} - 3 x + 1\) that satisfies the Mean Value Theorem on \(\displaystyle [-6, 4]\).
The slope of the secant line is \(\displaystyle m = \frac{f(4) -f(-6)}{4-(-6)}=1\). Setting the derivative equal to the slope of the secant line gives the equation \(\displaystyle 1 = - 4 x - 3\). The solution is \(\displaystyle c = -1\).
\begin{question}Find the value of $c$ for $f(x)=- 2 x^{2} - 3 x + 1$ that satisfies the Mean Value Theorem on $[-6, 4]$.
\soln{9cm}{The slope of the secant line is $m = \frac{f(4) -f(-6)}{4-(-6)}=1$. Setting the derivative equal to the slope of the secant line gives the equation $1 = - 4 x - 3$. The solution is $c = -1$. }
\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)=- 2 x^{2} - 3 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%202%20x%5E%7B2%7D%20-%203%20x%20%2B%201%20" alt="LaTeX: \displaystyle f(x)=- 2 x^{2} - 3 x + 1 " data-equation-content=" \displaystyle f(x)=- 2 x^{2} - 3 x + 1 " /> that satisfies the Mean Value Theorem on <img class="equation_image" title=" \displaystyle [-6, 4] " src="/equation_images/%20%5Cdisplaystyle%20%5B-6%2C%204%5D%20" alt="LaTeX: \displaystyle [-6, 4] " data-equation-content=" \displaystyle [-6, 4] " /> . </p> </p><p> <p>The slope of the secant line is <img class="equation_image" title=" \displaystyle m = \frac{f(4) -f(-6)}{4-(-6)}=1 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7Bf%284%29%20-f%28-6%29%7D%7B4-%28-6%29%7D%3D1%20" alt="LaTeX: \displaystyle m = \frac{f(4) -f(-6)}{4-(-6)}=1 " data-equation-content=" \displaystyle m = \frac{f(4) -f(-6)}{4-(-6)}=1 " /> . Setting the derivative equal to the slope of the secant line gives the equation <img class="equation_image" title=" \displaystyle 1 = - 4 x - 3 " src="/equation_images/%20%5Cdisplaystyle%201%20%3D%20-%204%20x%20-%203%20" alt="LaTeX: \displaystyle 1 = - 4 x - 3 " data-equation-content=" \displaystyle 1 = - 4 x - 3 " /> . The solution is <img class="equation_image" title=" \displaystyle c = -1 " src="/equation_images/%20%5Cdisplaystyle%20c%20%3D%20-1%20" alt="LaTeX: \displaystyle c = -1 " data-equation-content=" \displaystyle c = -1 " /> . </p> </p>