why does the jacobian point towards the maxima of a function?

Asked Jun 26 '19 at 20:45

Active Jun 29 '19 at 08:13

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I can derive the Jacobian of a multivariate function $f(x,y)$ by creating a horizontal vector: $$\left[\frac{∂f}{∂x} ~,~ \frac{∂f}{∂y}\right].$$

This vector will point to the maximum value of the function.

But why is this? Why does arranging the partial derivatives of $f(x,y)$ into a vector mean that it points to the peak of the function? The partial derivative can tell us the rate of change in a parameter. I don't see how this connects to the function's maxima.

edited Jun 29 '19 at 08:13

nmasanta

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asked Jun 26 '19 at 20:45

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This may help: Why is gradient the direction of steepest ascent?. Or see here. – Minus One-Twelfth Jun 26 '19 at 20:48
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It does not point to the maximum value of the function, except in some special cases. The function might not have any maxima at all. It points in the direction in which the function increases the fastest at the given point, i.e., the direction of steepest ascent. – amd Jun 26 '19 at 21:18
A function could have a maximum value that it reaches at two different points. You could be looking at a point somewhere between those two points. The Jacobian cannot point at both of the maximum points! – David K Jun 29 '19 at 10:09

why does the jacobian point towards the maxima of a function?

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