how do I find point of inflection of any function such as y=3x^2+2x+1

Hiya :) A point of inflection is a point where it suddenly changes from a concave upward curve to a concave downward curve, or vice versa.

But that doesn't really help us much, does it? :D Here's a tip... POSSIBLE points of inflection are points where the second derivative is zero...

Is it really zero at second derivative? I didnt know that

No... But what I'm saying is... only points where the second derivative is zero could POSSIBLY be points of inflection. Now, what's the second derivative of this function?

If i have to find answer for the function i wrote in question, how would u do that Can u show me one example

Well, let's take the function x^3... It's second derivative is 6x, right?

yes

When is 6x = 0?

x=0

That's right. And no others, right?

no

Okay, so x=0 (and y=0, since y = x^3) is a POSSIBLE point of inflection. In fact, it IS a point of inflection, but more on that later. Now, back to your function, what is it second derivative?

6

Yeap :) Now ask yourself, when is this equal to zero?

no where

so no point of inflection right? :D

critical points are at 0 OR undefined.

That's right :) that means there are no POSSIBLE points of inflection, and therefore, no point of inflection. Makes sense, right? This has a graph in the shape of a parabola, which has a consistent concavity, be it upwards or downwards (upwards in this case) And in both cases, the concavity doesn't change... at all :)

okay that helps thank you :)

No problem :)

so mostly its gonna be for function whose exponents are > or = 3

Or functions which aren't polynomials at all ;)

Like the sine function... it has (infinitely) many points of inflection :D

yeah right

^sounds sarcastic XD

:D

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