alleryWhich statement about $f(x)=x^3-3x^2+4x$ on $\mathbf{R}$ is correct?
AIt is increasing on all of $\mathbf{R}$
BIt is decreasing on all of $\mathbf{R}$
CIt is increasing only for $x>1$
DIt is decreasing only for $x<1$
Answer & Solution
Correct answer: A. It is increasing on all of $\mathbf{R}$
1. $f'(x)=3x^2-6x+4$.
2. Complete the square: $3(x^2-2x+1)+1=3(x-1)^2+1$.
3. Since $3(x-1)^2\ge 0$, $f'(x)\ge 1>0$ for all $x$.
4. So $f$ is increasing on all of $\mathbf{R}$; the partial-interval options are wrong because $f'$ never becomes negative.
_Source: syllabus Mathematics Ch 6 "Application of Derivatives", p.7_
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