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Questions: Algebra BusinessCalculus
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Find the local maximum and minimum of \(\displaystyle f(x) = - x^{3} - 3 x^{2} - 3 x - 4\).
To find the critical numbers solve \(\displaystyle f'(x) = 0\). The derivative is \(\displaystyle f'(x) = - 3 x^{2} - 6 x - 3\). Solving \(\displaystyle - 3 x^{2} - 6 x - 3 = 0\) gives \(\displaystyle x = \left[ -1\right]\). Using the 2nd derivative test gives:
\(\displaystyle f''\left( -1 \right) = 0 \) The 2nd derivative test fails.
\begin{question}Find the local maximum and minimum of $f(x) = - x^{3} - 3 x^{2} - 3 x - 4$.
\soln{9cm}{To find the critical numbers solve $f'(x) = 0$. The derivative is $f'(x) = - 3 x^{2} - 6 x - 3$. Solving $- 3 x^{2} - 6 x - 3 = 0$ gives $x = \left[ -1\right]$. Using the 2nd derivative test gives:\newline $f''\left( -1 \right) = 0 $ The 2nd derivative test fails. }
\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 local maximum and minimum of <img class="equation_image" title=" \displaystyle f(x) = - x^{3} - 3 x^{2} - 3 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20-%203%20x%20-%204%20" alt="LaTeX: \displaystyle f(x) = - x^{3} - 3 x^{2} - 3 x - 4 " data-equation-content=" \displaystyle f(x) = - x^{3} - 3 x^{2} - 3 x - 4 " /> . </p> </p><p> <p>To find the critical numbers solve <img class="equation_image" title=" \displaystyle f'(x) = 0 " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20%3D%200%20" alt="LaTeX: \displaystyle f'(x) = 0 " data-equation-content=" \displaystyle f'(x) = 0 " /> . The derivative is <img class="equation_image" title=" \displaystyle f'(x) = - 3 x^{2} - 6 x - 3 " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20%3D%20-%203%20x%5E%7B2%7D%20-%206%20x%20-%203%20" alt="LaTeX: \displaystyle f'(x) = - 3 x^{2} - 6 x - 3 " data-equation-content=" \displaystyle f'(x) = - 3 x^{2} - 6 x - 3 " /> . Solving <img class="equation_image" title=" \displaystyle - 3 x^{2} - 6 x - 3 = 0 " src="/equation_images/%20%5Cdisplaystyle%20-%203%20x%5E%7B2%7D%20-%206%20x%20-%203%20%3D%200%20" alt="LaTeX: \displaystyle - 3 x^{2} - 6 x - 3 = 0 " data-equation-content=" \displaystyle - 3 x^{2} - 6 x - 3 = 0 " /> gives <img class="equation_image" title=" \displaystyle x = \left[ -1\right] " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20%5Cleft%5B%20-1%5Cright%5D%20" alt="LaTeX: \displaystyle x = \left[ -1\right] " data-equation-content=" \displaystyle x = \left[ -1\right] " /> . Using the 2nd derivative test gives:<br> <img class="equation_image" title=" \displaystyle f''\left( -1 \right) = 0 " src="/equation_images/%20%5Cdisplaystyle%20f%27%27%5Cleft%28%20-1%20%5Cright%29%20%3D%200%20%20" alt="LaTeX: \displaystyle f''\left( -1 \right) = 0 " data-equation-content=" \displaystyle f''\left( -1 \right) = 0 " /> The 2nd derivative test fails. </p> </p>