Doubt in newton method

The instructor said this line

So in other words, if I let F of X be the derivative of G of X, then by finding the zeros of F of X, I am minimising G prime of X.

I do not agree with it, f(x) = g'(x) then if f(x) = 0 it means derivative of g(x) is also zero, which is not only the case in minima, but also maxima.

Hi @tbhaxor ,

You are correct. You can’t say it is maxima or minima just by looking at zeros of its derivative. That’s why we use Newton’s method of Optimization. Which confirms its minima. But how? The answer is simple, minima has double derivative positive whereas maxima has double derivative negative. Derivative of f(x) means double derivative of g(x), right?

Hope your doubt is cleared.

Harsh Der.

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Makes sense if the double derivative is positive that means function is uplifting from its minimum point.

I have a follow up question, please give a graphical representation if possible

But the derivative is the tangent line, which is horizontal to x axis when its minima or maxima. Since its horizontal so no slope, shouldnt double derivative be zero, I mean “always”. Keeping analytical method aside, thinking graphically.

Hi @tbhaxor ,

Lets explore this question. Your question is valid.

Now answer this question. What is difference between minima and maxima?

Its simple, the left slope of a maxima is positive whereas right slope is negative. Which indicates slope was decreasing until maxima and then started decreasing. Now what is slope or derivative? It is rate of change of y with respect to x. Now see why double derivative is negative at maxima. Because the slope was decreasing. Whereas in case of minima the slope is increasing so its double derivative is positive.

Hope your doubt is cleared

Harsh Der.