Calculus help. How do you find points of inflection?

Question

and why do I need to use a sign chart? I'm only in calculus one, so if you have any nifty tricks that get me the answer easier, please just give me the long way, otherwise my teacher will mark my answers wrong on my test coming up because he hasn't taught us it yet.

Answer 1

There are two considerations for finding inflection points. We say (a,f(a)) is an inflection point of y = f(x) if and only if f"(a) = 0 AND f"(x) changes sign as x passes through x = a. The second part of the definition tells you why you need a sign chart.

It is easy to see that there are functions which have no inflection points, but have values where the second derivative is zero. For example, consider f(x) = x^4. The graph looks like a big letter U with a vertex at (0,0). Here f"(0) = 0. but there is no inflection at x = 0.

Answer 2

to get find a point of inflection you need to find the derivative of your function and equate it to 0
then solve to find the answer (1)

then you put the answer (1) +1 into the derivative which should be either + or -

then you put the answer (1) -1 into the derivative which should be the same sign as the answer before

example

f'(x)=x^2-1=0
x=1

so
f'(1.1)=(1.1)^2-1= (+ sign (no need for the number) )
f'(0.9)=(0.9)^2-1= (- sign (no need for the number) )

in this case the point at 1 was a maximum
but if both values came out as the same sign (+ or -)
then this would be a point of inlflexion

hope this helped

Answer 3

What I would do is find the second derivative of the function and then just set it equal to zero and solve for x.

Ex.
y=x^3-x+5
y'=3x^2-1
y''=6x
0=6x
x=0=point of inflection

We are studying this right now as well and I am 95% sure my way will work as well but you might want to ask your teacher if my way works just to be safe.