13. Let f(x, y) = 3x^3 - (y^2)/(2) + xy. Find the critical...

13. Let $f(x, y) = 3x^3 - \frac{y^2}{2} + xy$. Find the critical points of the function $f$. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point.

Transcript

00:01 Hello, let's have a look at the question.

00:02 So we have been given a function fxy which is equals to xy and e to the power minus x minus y.

00:10 So now first of all, let us calculate fx, that is the differentiation of f with respect to x.

00:18 So here we will get y into e to the power minus x minus y minus e to the power minus x minus y into e to the power minus x minus y into x.

00:29 Similarly, let us find fy, that is the differentiation of f with respect to y.

00:35 So this will be equals to x into e to the power minus x minus y minus e to the power minus x minus y into y.

00:45 Now to find the critical points, let us put f x is equal to zero.

00:51 That means y into e to the power minus x minus y minus e to the power minus x minus y minus e to the is equal to 0.

01:00 So from here we can say that y is equal to 0 or e to the power minus x minus y minus e to the power minus x minus y into x is equal to 0.

01:13 So now also we can put f y is equal to 0.

01:18 So from here we will get that x into e to the power minus x minus x minus e to the power minus x minus y into y is equal to zero from here we will get x is equals to zero or we will get e to the power minus x minus y minus e to the power minus x minus y into y is equals to zero from these values we can say e to the power minus x minus y minus e to the power minus x minus y into x is equal to so from here we get x is equal to e to the power minus x minus y upon e to the power minus x minus y which is equals to one.

01:59 Similarly for the second we have e to the power minus x minus y minus e to the power minus x minus y into y is equal to zero.

02:09 So from here also we will get y is equals to one.

02:13 So the critical point that we got for these are 0 comma 0 and so there are two critical points here.

02:22 Now let us find the nature of these critical points so that we will apply now second derivative test.

02:30 So let us calculate f xx that is equals to y into minus two e to the power minus y minus x plus e to the power minus y minus x into x.

02:43 Now similarly let us calculate f y y y...

Question

13. Let $f(x, y) = 3x^3 - \frac{y^2}{2} + xy$. Find the critical points of the function $f$. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point.

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