# Chapter 3 - Applications of Differentiation - 3.1 Exercises: 14

$f(x)$ has two critical numbers: {$-1, 1$}.

#### Work Step by Step

Using the quotient rule: $f’(x)=(\frac{u(x)}{v(x)})'=\frac{u'(x)v(x)-v'(x)u(x)}{(v(x))^2}$ $u(x)=4x; u'(x)=4$ $v(x)=x^2+1; v'(x)=2x$ $f'(x)=\dfrac{4(x^2+1)-(2x)(4x)}{(x^2+1)^2}=\dfrac{4(1-x^2)}{(x^2+1)^2}.$ $f'(x)=0\to4(1-x^2)=0\to (1-x)(1+x)=0\to x=1$ or $x=-1.$ $f'(x)$ is defined for all values since the denominator is never $0.$ $f(x)$ has two critical numbers: {$-1, 1$}.

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