Find critical points (minima) from gradient (2D vector)

Question

I'm trying to find critical points (minima) for a gradient in 2D space. The following are partial derivative and its polynomial equation:

To find critical points I will need to set gradient to 0 and solve for 2 vectors span across $x_1$ and $x_2$. Could anyone provide steps in finding critical point for multi-variable equations? I'm trying to self-study and having this problem solved as a reference will be very helpful to my learning progress.

Thank you!

Answer 1

Setting the gradient of a multi variable equation equal to zero will give you the (local)minimum, (local)maximum or saddle points.

In your case, even though the gradient equation system is non-linear, we note that adding the two equations we get $$8x_1 + 8x_2 = 0 \Rightarrow x_1=-x_2.$$

Now we just substitute it in either the first or the second equation of the gradient system. Doing it for the first equation yields $$-4(x_2+x_2)^3+8x_2-1=0\Rightarrow-32x_2^3+8x_2-1=0$$ Solving it (using numerical methods, for example) and using the fact that $x_1 = -x_2$, we get the following points: $$(-0.13480, 0.13480), \quad (-0.41878, 0.41878), \quad (0.55358, -0.55358).$$

This is the contour plot of $F(x)$

Analyzing this plot (or doing it numerically) we conclude that:

• $(-0.13480, 0.13480)$: saddle point;
• $(-0.41878, 0.41878)$: local minimum;
• $(0.55358, -0.55358)$: global minimum.