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!