Find the maximum value of x 2y 5z as x y z varies among...

Find the maximum value of -x + 2y + 5z as (x, y, z) varies among points in {(x, y, z) in R^3: ||(x, y, z)|| = 1}. 1. Find the maximum value of x + 2y + 5z as (x, y, z) varies among points in {(x, y, z) in R^3: ||(x, y, z)|| = 1}.

Transcript

00:01 Hi, in this problem we are given f of x, y, z equal to x square plus y square plus y square plus z square, subject to x square plus y square equal to 16.

00:10 Consider this as equation number a, and we are required to find out the maximum and the minimum value.

00:15 So first let us consider that g of x is equal to x square plus y square plus z square minus 16.

00:27 So now we will find out the critical using lagrange's multipliers.

00:33 Therefore, let us consider delta f of xy z equal to lambda delta g of x y z.

00:45 So now for f of x equal to lambda g of x, we get 2xy square z square equal to 2 lambda x consider this as equation number one again for f of y equal to lambda g of y we have 2 x square y z square equal to 2 lambda y consider this as equation number two and again for f of z equal to lambda g of z we have 2 x square y square x equal to 2 lambda z consider this as equation number 3 now let us multiply x y and z in equation 1 equation 2 and equation 3 therefore we have 2 x square y square z square equal to 2 lambda x square and 2 x square y square x square equal to 2 lambda y square and 2 x square y square z square equal to 2 lambda z square so therefore further simplifying this implies that x square is equal to y square is equal to z square consider this as equation number four further substituting this value in equation number a we have x square plus x square plus x square equal to 16 and further simplifying we get x equal to plus minus 4 divided by square root 3 and this also implies that y equal to z is equal to plus minus 4 divided by square root 3.

02:38 Therefore the critical points are x y z equal to plus minus 4 divided by square root 3 and plus minus 4...

Question

Find the maximum value of -x + 2y + 5z as (x, y, z) varies among points in {(x, y, z) in R^3: ||(x, y, z)|| = 1}. 1. Find the maximum value of x + 2y + 5z as (x, y, z) varies among points in {(x, y, z) in R^3: ||(x, y, z)|| = 1}.

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