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I've been improving my skills at solving these simplification problems. Let's find the extrema of the function:
\n" ); document.write( "$$f(x)=\frac{x}{x^2+2}$$
\n" ); document.write( "On the interval $[-1, 4]$.\r
\n" ); document.write( "\n" ); document.write( "We can find the extrema of a function by finding its critical points, which are the points where the derivative is zero or undefined.\r
\n" ); document.write( "\n" ); document.write( "**Steps to solve:**
\n" ); document.write( "**1. Differentiate the function:**
\n" ); document.write( "$$f'(x)=\frac{2-x^2}{(x^2+2)^2}$$
\n" ); document.write( "**2. Set the derivative equal to zero and solve for x:**
\n" ); document.write( "$$f'(x)=0$$
\n" ); document.write( "$$\frac{2-x^2}{(x^2+2)^2}=0$$
\n" ); document.write( "$$2-x^2=0$$
\n" ); document.write( "$$x=\pm\sqrt{2}$$
\n" ); document.write( "**3. Evaluate the function at the critical points and endpoints of the interval:**
\n" ); document.write( "$$f(-1)=\frac{-1}{(-1)^2+2}=-\frac{1}{3}$$
\n" ); document.write( "$$f(\sqrt{2})=\frac{\sqrt{2}}{(\sqrt{2})^2+2}=\frac{\sqrt{2}}{4}$$
\n" ); document.write( "$$f(4)=\frac{4}{4^2+2}=\frac{2}{10}$$
\n" ); document.write( "**4. Compare the values of the function at the critical points and endpoints to determine the extrema:**
\n" ); document.write( "The maximum value of the function is $\frac{\sqrt{2}}{4}$ at $x=\sqrt{2}$.
\n" ); document.write( "The minimum value of the function is $-\frac{1}{3}$ at $x=-1$.\r
\n" ); document.write( "\n" ); document.write( "**Answer:**
\n" ); document.write( "The extrema of the function are:
\n" ); document.write( "* Maximum value: $\frac{\sqrt{2}}{4}$ at $x=\sqrt{2}$
\n" ); document.write( "* Minimum value: $-\frac{1}{3}$ at $x=-1$
\n" ); document.write( " \n" ); document.write( "