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Questions: Algebra BusinessCalculus
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Find the value of \(\displaystyle c\) for \(\displaystyle f(x)=4 x^{2} + 5 x - 2\) that satisfies the Mean Value Theorem on \(\displaystyle [-9, 1]\).
The slope of the secant line is \(\displaystyle m = \frac{f(1) -f(-9)}{1-(-9)}=-27\). Setting the derivative equal to the slope of the secant line gives the equation \(\displaystyle -27 = 8 x + 5\). The solution is \(\displaystyle c = -4\).
\begin{question}Find the value of $c$ for $f(x)=4 x^{2} + 5 x - 2$ that satisfies the Mean Value Theorem on $[-9, 1]$.
\soln{9cm}{The slope of the secant line is $m = \frac{f(1) -f(-9)}{1-(-9)}=-27$. Setting the derivative equal to the slope of the secant line gives the equation $-27 = 8 x + 5$. The solution is $c = -4$. }
\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)=4 x^{2} + 5 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D4%20x%5E%7B2%7D%20%2B%205%20x%20-%202%20" alt="LaTeX: \displaystyle f(x)=4 x^{2} + 5 x - 2 " data-equation-content=" \displaystyle f(x)=4 x^{2} + 5 x - 2 " /> that satisfies the Mean Value Theorem on <img class="equation_image" title=" \displaystyle [-9, 1] " src="/equation_images/%20%5Cdisplaystyle%20%5B-9%2C%201%5D%20" alt="LaTeX: \displaystyle [-9, 1] " data-equation-content=" \displaystyle [-9, 1] " /> . </p> </p><p> <p>The slope of the secant line is <img class="equation_image" title=" \displaystyle m = \frac{f(1) -f(-9)}{1-(-9)}=-27 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7Bf%281%29%20-f%28-9%29%7D%7B1-%28-9%29%7D%3D-27%20" alt="LaTeX: \displaystyle m = \frac{f(1) -f(-9)}{1-(-9)}=-27 " data-equation-content=" \displaystyle m = \frac{f(1) -f(-9)}{1-(-9)}=-27 " /> . Setting the derivative equal to the slope of the secant line gives the equation <img class="equation_image" title=" \displaystyle -27 = 8 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20-27%20%3D%208%20x%20%2B%205%20" alt="LaTeX: \displaystyle -27 = 8 x + 5 " data-equation-content=" \displaystyle -27 = 8 x + 5 " /> . The solution is <img class="equation_image" title=" \displaystyle c = -4 " src="/equation_images/%20%5Cdisplaystyle%20c%20%3D%20-4%20" alt="LaTeX: \displaystyle c = -4 " data-equation-content=" \displaystyle c = -4 " /> . </p> </p>