Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)f(x, y) = 3 − 2x + 4y − x2 − 4y2local maximum value(s)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;local minimum value(s)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;saddle point(s)&nbsp;&nbsp;&nbsp;&nbsp;(x, y, f)&nbsp;&nbsp;

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Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)f(x, y) = 3 − 2x + 4y − x2 − 4y2local maximum value(s)&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;local minimum value(s)&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;saddle point(s)&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;(x, y, f)&amp;nbsp;&amp;nbsp;

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To find the local maximum and minimum values and saddle point(s) of the functionf(x,y)=3−2x+4y−x2−4y2we need to find the critical points and determine their nature using the second partial derivative test.First, we take the partial derivatives of f with respect to x and y:∂f∂x=−2x−2and∂f∂y=4−8yNext, we set these partial derivatives equal to zero and solve for x and y:−2x−2=0⇒x=−14−8y=0⇒y=12Therefore, the only critical point of f is (−1,12).To determine the nature of this critical point, we need to compute the second partial derivatives of f:∂2f∂x2=−2∂2f∂x∂y=0∂2f∂y2=−8At the critical point (−1,12), we have:D=∂2f∂x2∂2f∂y2−(∂2f∂x∂y)2=(−2)(−8)−(0)2=16&amp;gt;0and∂2f∂x2=−2&amp;lt;0Therefore, the critical point (−1,12) is a local maximum of f.

Find the local maximum and minimum values and

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New Questions in Algebra I

local maximum value(s)
local minimum value(s)
saddle point(s)	(x, y, f)