Answer to Question #247307 in Calculus for Susan

Question #247307

Given the function f(x,y) = e^x ( x^2 - y^2) find all critical points and classify each as a local maximum, local minimum, or neither.


1
Expert's answer
2021-10-06T17:44:44-0400

Solution.

Find the partial derivatives:

"\\frac{\\partial f}{\\partial x}=2xe^x+(x^2-y^2)e^x,"

"\\frac{\\partial f}{\\partial y}=-2ye^x."

Let's solve the system of equations"2xe^x+(x^2-y^2)e^x=0,\\newline\n-2ye^x=0."

From the second equation "y=0." Substitute "y=0" in the first equation of the system and find x:

"(2x+x^2)e^x=0,\\newline\n2x+x^2=0,\\newline\nx(2+x)=0,\\newline\nx_1=0, x_2=-2."

So we have two critical points (0,0) and (-2,0).

Find the partial derivatives of the second order:

"B=\\frac{\\partial ^2f}{\\partial x\\partial y}=-2ye^x,"

"A=\\frac{\\partial ^2f}{\\partial x^2}=4xe^x+(x^2-y^2)e^x+2e^x,"

"C=\\frac{\\partial ^2f}{\\partial y^2}=-2e^x."

Let us calculate the value of these second-order partial derivatives at the critical points.

Calculate the values ​​for point (0; 0):

"A=2, B=0, C=-2."

Calculate AC - B 2 = -4 <0, then there is no global extremum.

Calculate the values ​​for point (-2; 0):

"A=-\\frac{2}{e^2},B=0, C=-\\frac{2}{e^2}."

Calculate AC - B 2 = 4 / e 4 > 0 and A <0, then at the point (-2; 0) there is a local maximum and "f_{max}" (-2; 0) = 4 / e 2


Answer. Two critical points (0,0) and (-2,0). Point (-2,0) is a point of local maximum.

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