Thread: Multivariable Calculus - Minimum value along a curve

3. Let g(x,y)=x²+y³.
(a) Find the directional derivative of g at the point (−3, 2) in the direction (3, 5).

(b) Determine the tangent line to the curve g(x, y) = g(4, −2) at the point (4, −2).
(c) Find the tangent plane to the surface z = g(x, y) at the point (−1, 1, 2).
(d) What is the minimum of g along the curve x² + y² = 4/9?






Hey there! Looking for help on part (d) but don't know if the other information is relevant so I just posted the whole question.
Any help is appreciated!

- Maedbh

You could eliminate the x^2 in g(x,y) and differentiate with respect to y but you are probably supposed to be using Lagrange multipliers. It's a very easy example. Check your notes. You can use the other method as a check.

You are asked to find the max or min of on the curve . Look at the two gradients: and . points in the direction of greatest increase of g so you can find a maximum value by moving in the direction of until it is 0. But, here, you can't move in that direction because you must stay on the curve f(x,y)= 4/9. What you can do is move along the curve in the direction of the projection of on the curve- again until that projection is 0. That happens when is perpendicular to the curve. And since is perpendicular to the curve f(x,y)= constant, you want those two gradients parallel- that is, one is a multiple of the other. You want for some constant . That is the "Lagrange multiplier" a tutor refers to. Surely, you seen that in class or your textbook.