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Chapter 3: Problem 61
Find the maximum value of \(f(x)=x^{3}-3 x\) on the set of all real numbers \(x\) satisfying \(x^{4}+36 \leq 13 x^{2}\). Explain your reasoning.
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In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{2 x^{2}}{x^{2}+4} $$
Writing Consider the function \(f(x)=\frac{2}{1+e^{1 / x}}\) (a) Use a graphing utility to graph \(f\). (b) Write a short paragraph explaining why the graph has a horizontal asymptote at \(y=1\) and why the function has a nonremovable discontinuity at \(x=0\).
In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ g(x)=\sin \left(\frac{x}{x-2}\right), \quad x>3 $$
(a) Let \(f(x)=x^{2}\) and \(g(x)=-x^{3}+x^{2}+3 x+2 .\) Then \(f(-1)=g(-1)\) and \(f(2)=g(2) .\) Show that there is at least one value \(c\) in the interval (-1,2) where the tangent line to \(f\) at \((c, f(c))\) is parallel to the tangent line to \(g\) at \((c, g(c)) .\) Identify \(c .\) (b) Let \(f\) and \(g\) be differentiable functions on \([a, b]\) where \(f(a)=g(a)\) and \(f(b)=g(b) .\) Show that there is at least one value \(c\) in the interval \((a, b)\) where the tangent line to \(f\) at \((c, f(c))\) is parallel to the tangent line to \(g\) at \((c, g(c))\).
In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x^{2}}{x^{2}+9} $$
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