I need some help with this. I've been struggling through this last chapter of my Calc III class, and I'm not sure how to do this (although, it doesn't seem like it should be difficult to do)
$$
\text{Find the maximum and minimum values of }f(x, y) = 4x + y\text{ on the ellipse } x^{2} + 49y^{2} = 1 \\
\text{Maximum = _____}\\
\text{Minimum = _____}
$$
I know that I can find the critical point by taking partial derivatives so
$$
\frac{\partial{f}}{\partial{x}} = 4 \\
\frac{\partial{f}}{\partial{y}} = 1
$$
Which gives us the critical point $(4,1)$.
Here's where I get stuck, I know that we can then determine max/min from the equation $D = f_{xx} * f_{yy} – f_{xy}^2$ and based on the value of $D$, we know whether it is a max, a min, or a "saddle" point. It doesn't seem to apply in this situation, because I have an ellipse that I have to use as a constraint.
What do I do next?
UPDATE:
Based on everyone's suggestion, I've been looking into Lagrange multipliers. I think I'm still stuck.
$$
\bigtriangledown{f} = \lambda \bigtriangledown{g} \\
f_x = 4 = \lambda 2 * x\\
f_y = 1 = \lambda 98 * y \\
\text{we need to solve by using ratios of derivatives to remove } \lambda\\
\frac{4}{1} = \frac{\lambda 2 x}{\lambda 98 y} \\
196 y = x\\
\text{Now we use our ellipse to solve}\\
(196y)^2 + 49y^2 = 1 \\
38445 y^2 = 1 \\
y = \sqrt{\frac{1}{38465}} = y_0\\
\text{plug y back into our ellipse} \\
x^2 + 49 (\frac{1}{38465}) = 1\\
x = \sqrt{1- \frac{49}{38465}} = x_0
$$
I'm feeling somewhat accomplished, but I have no idea if what I'm doing is correct, or what to do with these new numbers
Best Answer
Lagrange multipliers do work for your problem, but that involves solving three simultaneous non-linear equations in three unknowns. There is another way that uses only one variable: parameterization.
Get a parameterization that describes the given curve in terms of only one variable. In the case of your ellipse you can use
$$x=\cos t, \quad y=\frac 17\sin t, \quad 0\le t\le 2\pi$$
Now you want to want to find the extrema of
$$\hat f(t)=4x+y=4\cos t+\frac 17\sin t$$
There are several ways to find the extrema of that, using calculus or just trigonometry. Here is a calculus way:
$$\hat f'(t)=-4\sin t+\frac 17\cos t=0$$ $$28\sin t=\cos t$$ $$\tan t=\frac 1{28}$$ $$\cos t=\sqrt{\frac 1{\tan^2 t+1}}=\pm\frac{28}{\sqrt{785}}$$ $$\sin t=\sqrt{1-\cos^2 t}=\pm\frac{1}{\sqrt{785}}$$ $$x=\cos t=\pm\frac{28}{\sqrt{785}}$$ $$y=\frac 17\sin t=\pm\frac{1}{7\sqrt{785}}$$
where $x$ and $y$ have the same sign. This means the maximum of $f$ is
$$4x+y=4\frac{28}{\sqrt{785}}+\frac{1}{7\sqrt{785}}=\frac{785}{7\sqrt{785}}=\frac{\sqrt{785}}{7}\approx 4.00255$$
and the minimum is the negative of that. This graph confirms the maximum.