For what values of a and b will the point A(1,3) become an inflection point of the curve f(x)=ax^3+bx^2?
This seems too easy and I'm surprised to only find one solution but here's what I did.
We know f(1)=3 so a+b=3
An inflection point is when the second derivative is zero (and changes sign on either side)
f'(x)=3ax^2+2bx
f"(x)=6ax+2b (this is linear so unless a=0 we have the sign change)
For the inflection to be at x=1
f"(1)=6a+2b=0
And we have a system of two equations and two unknowns. Solving gives a=-1.5, b=4.5
Double checking the graph f(x)=-1.5x^3 + 4.5x^2 works.
Thinking about this, maybe it shouldn't be surprising there is only one solution since the cubic has only been given two degrees of freedom. If it had another variable term, we'd probably get a family of solutions.
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Posted by Jer
on 2019-03-26 15:59:24 |
solution