00:01
For this problem, we're asked to find the points on the sphere, x squared plus y squared plus z squared equals 25, where f of x, y, z equals x plus 2y plus 3z, has its maximum and minimum values.
00:12
To begin, we set up our lagrange multiplier equations, which will give us that 1 must be equal to 2 times lambda times x, 2 must be equal to 2 times lambda times y, which prominently gives us that lambda must be equal to 1 over y, and we have that 3 must be equal to 2 times lambda times z so we then have that lambda must be equal to 3 over 2 z which in turn must also looking at our first equation we have that lambda must be equal to 1 over 2x so 3 over 2 z must be equal to 1 over 2 x which gives us that z over 3 must be equal to x, so z equals 3x, and we have that 1 over y must be equal to 1 over 2x, so y must be equal to 2x.
01:09
Now we can substitute these definitions for y and z into the equation of our sphere.
01:16
So we'd have, let's see here, x squared, plus 2x all squared, so x squared plus 4x squared, plus 9x squared, must be equal to 25...