[tex]\bf f(x)=\cfrac{x}{x^2+2}\implies \cfrac{dy}{dx}=\cfrac{1(x^2+2)-(x)(2x)}{(x^2+2)^2}\\\\\\ \cfrac{dy}{dx}=\cfrac{x^2+2-2x^2}{(x^2+2)^2}
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\cfrac{dy}{dx}=\cfrac{2-x^2}{(x^2+2)^2}\implies 0=2-x^2\implies x=\pm\sqrt{2}
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\textit{now }-\sqrt{2}\textit{ is outside the range of }[0,4],\textit{ so is only }\sqrt{2}[/tex]
the denominator yields no critical points, so is only that one, which IS within the range of [0, 4].
f(0) = 0 and f(4) is about 0.2222... whilst f(√2) is about 0.3536
now, doing a first-derivative test, the √2 points is a maximum, and and the 0 and 4 are both minima, from which the 0 is lowest than the 4, notice f(0) = 0 and f(4) is up above that.
so the absolute minimum is f(0), and the absolute maximum is f(√2).
