Maxima and Minima (vector calculus)

[Win_odd Dhamnekar]

Hi,Hi,

 

[BigBeachBanana]

Hi,Hi,

View attachment 32110

Can you even tell where the unit circle is? The keyword is "might". Not that it will have it. You need to show it mathematically.

1) See Wolfram

2) You can use the free online GeoGebra 3-D Graphing.

 

[blamocur]

The graph has rotational symmetry around the origin in the XY plane, which means that [imath]f[/imath] depends only on the radius and you can reduce the problem to a 1D function. This would make it slightly easier to plot and analyze.

 

[Win_odd Dhamnekar]

1)Following answer was obtained by me on another math help website which I reproduce here.

Wolfram


If you expand Taylor-series, you can immediately see it's a maximum, because there is no 1st-order correction and the 2nd-order correction is negative.

2) Following is the second answer, I got on internet , which I reproduce below:


The easiest way to visualise this since the circle is symmetric is to consider the simpler function

f(p) = p*e^-p, where p = (x² + y²)

any plane section will have this form.

When p = 0.5 f(p) is approximately 0.3

When p =1, f(p) is approximately 0.4

When p = 2 f(p) is approximately 0.2

and f(p) is continuous so there is a max between p = (x² + y²) = 0.3 and p = (x² + y²) = 2