If f(2)=3, f'(2)=1, and f''(2)=0, is f(2,3) an inflection point? Why or why not?

It possibly could be.  It all depends on whether the function is concave up on one side of 2 and concave down on the other side of 2. If that is the case, it is an inflection point.
From what is given, we don't know if it is concave up or concave down on either side.

It could be y = x+1, for then f(2) = 3, f'(x) = 1, so f'(2) = 1, and f"(x) = 0, so f"(2) = 0.
In this case, neither side if concave up or down.

If we took f"(x) = 2-x, that would make f"(2) = 2.
That would make f'(x) = 2x - (x^2)/2 + C.
Since f'(2) = 1 and we can see that f'(2) = 2 + C, C is -1.
This makes f'(x) = 2x - (x^2)/2 - 1.

From there f(x) = x^2 - (x^3)/6 - x + C.
Since f(2) = 3 and f(2) can be seen to be 4 -4/3 - 2 + C = 14/3 + C,
that means C = 1/3, so f(x) = x^2 - (x^3)/6 - x + 1/3.

As can be seen, this is concave up on the left and concave down on the right.  


All Answers

Answers by Expert:

Ask Experts




Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology (reproduction, insusion of chemicals into bloodstream).


Experience in the area: I have tutored students in all areas of mathematics since 1980. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

Maybe not a publication, but I have respond to well oveer 8,500 questions on the PC. Well over 2,000 of them have been in calculus.

I aquired well over 40 hours of upper division courses. This was well over the number that were required. I graduated with honors in both my BS and MS degree from Oregon State University. I was allowed to jump into a few courses at college a year early.

Awards and Honors
I have been nominated as the expert of the month several times. All of my scores right now are at least a 9.8 average (out of 10).

Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.

©2016 All rights reserved.