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Find Critical Points and Extrema for Cubic and Rational Functions
Mathematics
Grade 11 (Senior High School)
Question Content
a) $f(x)=\\frac{1}{3}(x^3 - 27x)$; b) $g(x)=\\frac{x^2}{x+3}$
Correct Answer
a) Critical points at x = 3, x = -3; local maximum at (-3, 54), local minimum at (3, -54); b) Critical point at x = 0, x = -6; local minimum at (0, 0), local maximum at (-6, -12)
Detailed Solution Steps
1
Step 1 (for f(x)): Find the first derivative using power rule: $f'(x)=\\frac{1}{3}(3x^2 -27)=x^2 -9$
2
Step 2 (for f(x)): Set $f'(x)=0$, solve $x^2-9=0$ → $x=3$ or $x=-3$. Use second derivative test: $f''(x)=2x$. $f''(-3)=-6<0$ (local max), $f''(3)=6>0$ (local min). Calculate $f(-3)=54$, $f(3)=-54$
3
Step 1 (for g(x)): Use quotient rule for derivative: $g'(x)=\\frac{2x(x+3)-x^2(1)}{(x+3)^2}=\\frac{x^2+6x}{(x+3)^2}$
4
Step 2 (for g(x)): Set $g'(x)=0$, solve $x^2+6x=0$ → $x=0$ or $x=-6$. Use second derivative test or sign chart: $g''(0)=\\frac{6}{9}>0$ (local min, $g(0)=0$), $g''(-6)=\\frac{-6}{9}<0$ (local max, $g(-6)=-12$)
Knowledge Points Involved
1
Power Rule for Differentiation
A rule for finding derivatives of monomials: $\\frac{d}{dx}[x^n]=nx^{n-1}$, used to differentiate polynomial functions like $f(x)$
2
Quotient Rule for Differentiation
A rule for derivatives of rational functions: $\\frac{d}{dx}[\\frac{u}{v}]=\\frac{u'v-uv'}{v^2}$, used when differentiating functions like $g(x)$ where one polynomial is divided by another
3
Second Derivative Test for Extrema
A method to classify critical points: if $f''(c)<0$, the function has a local maximum at $x=c$; if $f''(c)>0$, the function has a local minimum at $x=c$
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