Answer to Question #247892 in Calculus for Susan

Question #247892

Given the function, f(x,y) = x^3 y + 12x^2 - 8y Find all critical points and classify each as a local maximum, local minimum, or neither.


1
Expert's answer
2021-10-12T04:41:08-0400

Given function is "f(x,y)=x^3y+12x^2-8y=0"

Differentiating the function w.r.t. "(x)," we get:

"3x^2y+x^3\\frac{dy}{dx}+24x-8\\frac{dy}{dx}=0\\\\\n\\Rightarrow \\frac{dy}{dx}=\\frac{24x+3x^2y}{8-x^3}"

For critical points, put "\\frac{dy}{dx}=0," we get:

"\\frac{24x+3x^2y}{8-x^3}=0\\\\\n\\Rightarrow 24x+3x^2y=0\\\\\n\\Rightarrow 3x(8+xy)=0"

"\\therefore x=0 \\ or \\ xy=-8"


Case 1: If "x=0" , then:

"f(x,y)=-8y=0\\\\\n\\Rightarrow y=0"

Case 2: If "xy=-8" , then:

"f(x,y)=-8x^2+12x^2-8y=0\\\\\n\\Rightarrow 4x^2-8y=0\\\\\n\\Rightarrow \\frac{x^2}{2}=y"

So, the given function can be written as:

"x^3(\\frac{x^2}{2})+12x^2-4x^2=0\\\\\n\\Rightarrow \\frac{x^5}{2}+8x^2=0\\\\\n\\Rightarrow x^2(\\frac{x^3}{2}+8)=0\\\\\n\\Rightarrow x=0 \\ or\\ x^3=-16\\\\\n\\Rightarrow x=0 \\ or\\ x=-(16)^{\\frac{1}{3}}"

"\\therefore y=0 \\ or\\ y=\\frac{(16)^{\\frac{2}{3}}}{2}"

So, from both cases, we obtain the same point: "(0,0)" and "(-(16)^{\\frac{1}{3}}, \\frac{(16)^{\\frac{2}{3}}}{2})"


Now, calculating double derivative, we get:

"6xy+3x^2\\frac{dy}{dx}+3x^2\\frac{dy}{dx}+x^3\\frac{d^2y}{dx^2}+24-8\\frac{d^2y}{dx^2}=0\\\\\n\\Rightarrow \\frac{d^2y}{dx^2}=\\frac{6xy+6x^2\\frac{dy}{dx}+24}{8-x^3}"

Now, calculating value of double derivative at "(0,0)" , we get:

"(\\frac{d^2y}{dx^2})_{(0,0)}=\\frac{24}{8}=3" which is positive. Hence, the point is local minimum.


Now, calculating value of double derivative at "(-(16)^{\\frac{1}{3}}, \\frac{(16)^{\\frac{2}{3}}}{2})", we get:

"(\\frac{d^2y}{dx^2})_{(-(16)^{\\frac{1}{3}}, \\frac{(16)^{\\frac{2}{3}}}{2})}=\\frac{-48+0+24}{8+16}=-1" which is negative. Hence, the point is local maximum.


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