Find the extreme values of f(x,y,z) = xyz subject to x^2 +...

Find the extreme values of f(x,y,z) = xyz subject to x^2 + y^2 + z^2 = 4 and x + y = 2. A. The maximum is ?3 at (1, -1, -?3). The minimum is -?3 at (1, -1, ?3). B. The maximum is ?2 at (1, 1, ?2). The minimum is -?2 at (1, 1, -?2). C. The maximum is ?2 at (1, -1, -?2). The minimum is -?2 at (1, -1, ?2). D. The maximum is ?3 at (1, 1, ?3). The minimum is -?3 at (1, 1, -?3).

Transcript

00:01 Here in this question we have to find out the extreme value of the point that is f of x y and z that is equals to x y z which are subjected to the constraint that is x raised to the power 2 plus y raised to the power 2 plus z raised to the power 2 that is equals to 4 and x plus y that is equals to 2 where g of x y and z that is equals to x raised to the power 2 plus y raised to the power 2 plus z raised to the power 2 minus 4 where h of x y and z is equals to x plus y minus 2.

00:42 So here we are using the lagrange multiplier method.

00:50 So from the lagrange multiplier method we are considering about the gradient of f that is equals to y z x z and x y and the gradient of g from here is 2 x 2 y 2 z and whereas the gradient of h is equals to 1 1 and 0.

01:14 So from here we can say that the gradient of f is equals to lambda of gradient of g plus mu of gradient of h.

01:22 In the same way the gradient of g is equals to sorry from here let's complete about the gradient of the f.

01:33 So gradient of f here what we have taken the value of the gradient of f is this so plugging into the value here in this equation.

01:40 So the gradient of f from here is y z x z and x y that from here is equals to lambda multiplied by the gradient of z that is 2 of x 2 of y 2 of z plus mu multiplied by the gradient of h that is 1 1 and 0.

01:59 So we get the value that y of z is equals to 2 lambda of x plus mu x of z is equals to 2 lambda y plus mu and x of y is equals to 2 lambda z.

02:13 Since x y is equals to 2 lambda z so we can say that x y divided by the 2 of z is equals to the lambda.

02:23 Now we are having the value of lambda so now we have to substitute its value that is lambda is x y divided by the 2 g.

02:34 Now we are considering about y z that is equals to 2 lambda x plus mu.

02:39 So substituting the value of lambda here so we can say that y z is equals to 2 multiplied by the lambda that is x y divided by the 2 z multiplied by the x plus mu.

02:53 So solving this term we get the value that is y of z is equals to the power 2 is x square of y plus z of mu.

03:01 So we can write it as that now from here simplifying this term to get the value of mu.

03:08 So mu from here is equals to y of z is equals to the power 2 minus x is equals to the power 2 of y divided by z...

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