Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. f(x, y)=sin⁡(2πx)cos⁡(πy), for |x|≤12 and |y|≤12

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Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. f(x, y)=sin⁡(2πx)cos⁡(πy), for |x|≤12 and |y|≤12

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Step 1We must determine the partial derivatives of the given function with respect to x and y in order to determine the critical pointfx(x,y)=2π(cos⁡2πx)cos⁡(πy)fy(x,y)=π(−sin⁡πy)sin⁡(2πx)Step 2Equating the derivatives to 0, to find the critical points2π(cos⁡2πx)cos⁡(πy)=0⇒cos⁡2πx=0⇒2πx=cos−1⁡(0)⇒2πx=π2⇒x=14cos⁡πy=0⇒πy=cos−1⁡(0)⇒πy=π2⇒y=12Therefore the critical points from this is&amp;nbsp;(14,a)&amp;nbsp;and&amp;nbsp;(b,12)π(−sin⁡πy)sin⁡(2πx)=0⇒−sin⁡πy=0⇒sin⁡πy=0⇒πy=sin−1⁡(0)⇒y=0sin⁡(2πx)=0⇒2πx=sin−1(0)⇒x=0The critical points from this are (0,c) and (d,0)Therefore the critical points of the given function are(14,a),&amp;nbsp;(b,12),&amp;nbsp;(0,x)&amp;nbsp;and (d,0)Step 3Let us use the second derivative test to determine whether the critical points correspond to a local maximum, a local minimum,or a saddle point.D(x,y)=f×(x,y)fyy(x,y)−(fxy(x,y))21. If&amp;nbsp;D(x,y)&amp;gt;0&amp;nbsp;and&amp;nbsp;f×(x,y)&amp;lt;0, then f has a local maximum value at (x,y)2. If&amp;nbsp;D(x,y)&amp;gt;0&amp;nbsp;and&amp;nbsp;f×(x,y)&amp;gt;0, then f has a local minimum value at (x,y)3. If&amp;nbsp;D(x,y)&amp;lt;0&amp;nbsp;then f has a saddle point at (x,y)4. If&amp;nbsp;D(x,y)=0&amp;nbsp;then the test is inconclusiveLet us find&amp;nbsp;f×(x,y),&amp;nbsp;fyy(x,y),&amp;nbsp;fxy(x,y)&amp;lt;

Find the critical points of the following functions.Use the Second Derivative Te

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