Finding Inflection Points: Solving for a,b, and c

[AdiV]

Hi, can someone help me guide me on how to find inflections? I am not too sure how to do it in such a problem as the follows:

Homework Statement

Determine a, b, and c so that the function
f(x) = x^3 + ax^2 + bx + c has critical points at (1,5) and an inflection point at (2,3)

Homework Equations

The Attempt at a Solution

I am not too sure how to pursue this, for critical points I know the derivative needs to equal zero and then you find the critical points that way, but I don't know what to do for the inflection.
Thank You!

 

[rocomath]

1st derivative - critical points

2nd derivative - inflection points

 

[AdiV]

Yep, thanks I figured it out
a = -9/2
b = 11

But I can't find out what c is;
This is wat I did by the way;

y = x3 + ax2 + bx + c

y' = 3x2 + 2ax + b

y'' = 6x + 2a

Take y' and plug in (1, 5)

5 = 3(1) + 2a(1) + b ==> 2a + b = 2 ... (1)

Take y'' and plug in (2, 3)

3 = 6(2) + 2a ==> a = -9/2 ... (2)

Then, plug (2) into (1) to find b

b = 11

 

[Rainbow Child]

... Take y' and plug in (1, 5) Ouppss!...

...Take y'' and plug in (2, 3) Ouppps, again!...

The points (1,5), (2,3) are (x,y) points not (x,y') or (x,y'') points!

On critical points y'(x_o)=0

On inflection points y''(x_o)=0

 

[AdiV]

Ohh, ok, yes, I fixed it now, I have my answer to be

a = -6
b = 9
c = 1

 

[Rainbow Child]

Ohh, ok, yes, I fixed it now, I have my answer to be

a = -6
b = 9
c = 1

And that's the correct one!