Find a polynomial function given local extrema

Question

I'm struggling to find a polynomial function with given local extrema.

For instance, consider the problem:

Find a function with a local maximum at $(0,0)$ and a local minimum at $(1,−5)$.

I start with $f(0)=0$, $f(1)=−5$, $f′(0)=0$ (or is it undefined?), $f′(1)=0$ and then try to solve $f(x)=ax^3+bx^2+cx+d$ with the above values.

So far I have $f(1)=-5$ gives $a+b+c+d=-5$ and $f'(1)=0$ gives $3a+2b+c=0$, but I'm unsure how to go about working with $f(0)$ and $f'(0)$.

Is it $f(0)=0$ so $a+b+c=d=0$ so $d=0$ and $f'(0)=a+b=c=0$ so $c=0$? This seems incorrect, but I'm not sure else I can do here.

Answer 1

How to proceed

as you mentioned $f(0)=0$, $f'(0)=0$, $f(1)=5$, $f'(1)=0$, because those are local minimas and maximas and therefore, there derivates are equal to $0$.

Second step now is to use $f(x)=a.x^3 + b.x^2 + c.x +d$ and use the values we previously talked about in this function

Replace for each equation the x by its value. You will end up having $d=c=0$

The rest will keep it as a challenge for you.

Answer 2

We have $f'(x)=kx(x-1)$, hence $f(x)=k(x^3/3-x^2/2)+c$. From $f(0)=0$ it follows that $c=0$. As $-5=f(1)=k(1/3-1/2)=-k/6$ we know $k=30$, hence $f(x)=10x^3-15x^2$.