Question

Solve the given optimization problem by using substitution. HINT [See Example 1.]

Find the maximum value of
f(x, y, z) = 5 − x^(2) − y^(2) − z^(2)
subject to z = 5y.

Answer 1

Take all the derivatives, with respect to each variable.

Answer 2

so how do you find the max?

Answer 3

Wait, does z = 5y simply mean to substitute z as 5y or some crazy thing they teach in real calc courses?

Answer 4

i don't know but i was thinking it means to substitute for it

Answer 5

Okay, well then first things first. Simplify.
$$f(x, y, z) = 5 - x^2 - 26y^2$$
I just graphed this surface, it's an elliptic paraboloid.
Now, take the derivatives with respect to and x and y, and solve. If there is a common x = y = 0, then that point is a maximum.

Answer 6

so fx= -2x
fy= -52y

Answer 7

yes those are the correct derivatives

Answer 8

so how do you find the max

Answer 9

oh wait, but isn't the max 5?

Answer 10

whatttt

Answer 11

oh yeah it is

Answer 12

yup,

Answer 13

what about the corresponding points x y z

Answer 14

and how did you know 5 was the max?

Answer 15

logically

Answer 16

5 is the only term that doesn't change with x^2 or y^2

Answer 17

*x or y

Answer 18

oh okay thank you!