# Find the max/min values of s^2+t^2 on a curve by the method of Lagrange Multipliers

• Oct 24th 2010, 03:46 PM
Runty
Find the max/min values of s^2+t^2 on a curve by the method of Lagrange Multipliers
I've gotten an answer for the first half of this, but not the second one, which I believe is some sort of trick question.

Find the maximum and minimum values of $s^2+t^2$ on the curve
$s^2+2t^2-2t=4$
by the method of Lagrange Multipliers.
Max: $(s,t)=(\pm 2, 1),s^2+t^2=5$
Min: $(s,t)=(0,2),s^2+t^2=4$, $(s,t)=(0,-1),s^2+t^2=1$

Now for the second half, which I believe is some sort of trick question.

If $s^2=4-2t^2+2t$ is substituted into $s^2+t^2$, we get a function $h(t)=4+2t-t^2$, which has only a maximum value on $R$. Explain how the extreme values obtained in the first part can be obtained from $h(t)$.
See, the issue is that $h(t)$ is a parabola, which has a maximum at 5, but no minimum (or at least no absolute minimum). So how would you use $h(t)$ to find the minimum values? That's what I feel is the "trick" part of the question.

If anyone could clarify the second half of the question, I'd appreciate it.
• Oct 25th 2010, 02:02 AM
Opalg
Quote:

Originally Posted by Runty
I've gotten an answer for the first half of this, but not the second one, which I believe is some sort of trick question.

Find the maximum and minimum values of $s^2+t^2$ on the curve
$s^2+2t^2-2t=4$
by the method of Lagrange Multipliers.
Max: $(s,t)=(\pm 2, 1),s^2+t^2=5$
Min: $(s,t)=(0,2),s^2+t^2=4$, $(s,t)=(0,-1),s^2+t^2=1$
If $s^2=4-2t^2+2t$ is substituted into $s^2+t^2$, we get a function $h(t)=4+2t-t^2$, which has only a maximum value on $R$. Explain how the extreme values obtained in the first part can be obtained from $h(t)$.
See, the issue is that $h(t)$ is a parabola, which has a maximum at 5, but no minimum (or at least no absolute minimum). So how would you use $h(t)$ to find the minimum values? That's what I feel is the "trick" part of the question.
The point is that h(t) is not defined for all values of t. The curve $s^2+2t^2-2t=4$ represents an ellipse with centre at (0,1/2) and minor axis 3/2. So t has to lie between –1 and +2. If you look for the minimum value of h(t) in that interval, you will find that it occurs at an endpoint.