What intervals is f(x)=5x2x2+4 concave up/down?

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anita1415snck · Accepted Answer

Step 1We will examine concavity by examining the sign of the second derivative.f(x)=5x2x2+4f′(x)=−40x(x2+4)2Finding f&quot;(x):fx)=−40(x2+4)2−(−40x)2(x2+4)(2x)(x2+4)4=(x2+4)[−40(x2+4)+160x2](x2+4)4=−40(x2+4)+160x2(x2+4)3Step 2fx)=120x2−160(x2+4)3Step 2A function can typically change sign by crossing the x-axis (being equal to 0) or being discontinuous (teleporting across the x axis).For real x in this scenario, f&quot;(x) is never discontinuous, andfx)=0&amp;nbsp;when&amp;nbsp;3x2−4=0,so&amp;nbsp;x=±23Test each interval after cutting the number line into intervals:(−∞,−23)fx)&amp;nbsp;is positive, so the graph of f is concave up(−23,23)fx)&amp;nbsp;is negative, so the graph of f is concave down(23,∞)fx)&amp;nbsp;is positive, so the graph of f is concave up

What intervals is f(x)=\frac{5x^{2}}{x^{2}+4} concave up/down?

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