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Questions: Algebra BusinessCalculus
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Find the critical numbers of \(\displaystyle f(x)=x^{3} - \frac{51 x^{2}}{2} + 216 x - 13\).
Taking the derivative gives \(\displaystyle f'(x)=x^{2} - 17 x + 72\). The critical numbers of the zeros of the derivative. Setting it equal to zero and solving gives \(\displaystyle x^{2} - 17 x + 72 = 0 \iff \left(x - 9\right) \left(x - 8\right)=0\). The critical numbers are: [8, 9].
\begin{question}Find the critical numbers of $f(x)=x^{3} - \frac{51 x^{2}}{2} + 216 x - 13$.
\soln{9cm}{Taking the derivative gives $f'(x)=x^{2} - 17 x + 72$. The critical numbers of the zeros of the derivative. Setting it equal to zero and solving gives $x^{2} - 17 x + 72 = 0 \iff \left(x - 9\right) \left(x - 8\right)=0$. The critical numbers are: [8, 9]. }
\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 critical numbers of <img class="equation_image" title=" \displaystyle f(x)=x^{3} - \frac{51 x^{2}}{2} + 216 x - 13 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3Dx%5E%7B3%7D%20-%20%5Cfrac%7B51%20x%5E%7B2%7D%7D%7B2%7D%20%2B%20216%20x%20-%2013%20" alt="LaTeX: \displaystyle f(x)=x^{3} - \frac{51 x^{2}}{2} + 216 x - 13 " data-equation-content=" \displaystyle f(x)=x^{3} - \frac{51 x^{2}}{2} + 216 x - 13 " /> . </p> </p><p> <p>Taking the derivative gives <img class="equation_image" title=" \displaystyle f'(x)=x^{2} - 17 x + 72 " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%3Dx%5E%7B2%7D%20-%2017%20x%20%2B%2072%20" alt="LaTeX: \displaystyle f'(x)=x^{2} - 17 x + 72 " data-equation-content=" \displaystyle f'(x)=x^{2} - 17 x + 72 " /> . The critical numbers of the zeros of the derivative. Setting it equal to zero and solving gives <img class="equation_image" title=" \displaystyle x^{2} - 17 x + 72 = 0 \iff \left(x - 9\right) \left(x - 8\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20-%2017%20x%20%2B%2072%20%3D%200%20%5Ciff%20%5Cleft%28x%20-%209%5Cright%29%20%5Cleft%28x%20-%208%5Cright%29%3D0%20" alt="LaTeX: \displaystyle x^{2} - 17 x + 72 = 0 \iff \left(x - 9\right) \left(x - 8\right)=0 " data-equation-content=" \displaystyle x^{2} - 17 x + 72 = 0 \iff \left(x - 9\right) \left(x - 8\right)=0 " /> . The critical numbers are: [8, 9]. </p> </p>