Determine polynomial function of degree 4.

Question

The graph of a polynomial function $f(x)$ of degree $4$ with real coefficients has a local maximum at $(-3|3)$ and a local minimum at $(1|0)$ and no other local extremal points. Determine the function. (It's from a math textbook, 11th grade, Bavaria - current homework for our Eileen).

Ansatz: $f(x) = ax^4+bx^3+cx^2+dx+e$ with $f'(x)=4ax^3+3bx^2+2cx+d$.

The conditions given are $f(-3)=3$, $f'(-3)=0$, $f(1)=0$, and $f'(1)=0$.

So there is one condition missing to determine all five parameters. I guess it's from ''no other local extremal points''. Any idea? How does the graph look like at $\pm \infty$?

Answer 1

$f$ is a polynomial of degree $4$ so $f'$ is a polynomial of degree $3$. Given the hypotheses, it has two roots $-3$ and $1$ of odd multiplicity (or it would not be local extrema). Then the last root has to be real as well,given that the polynomial under study has real coefficients, contradicting the last hypothesis.

Answer 2

There is no such polynomial. You know that if $x$ is smaller than $-3$ and close to it, then $p(x)<p(-3)(=3)$. And, if the coefficient of $x^4$ is greater than $0$, $\lim_{x\to-\infty}p(x)=+\infty$. But then the restriction of $p(x)$ to $(-\infty,-3)$ has to have an absolute minimum, which will be a local minimum of $p(x)$. And if the coefficient of $x^4$ is smaller than $0$, you can apply the same argument to the point $x=1$.

Answer 3

If $a >0$, then $f$ approaches $+\infty$ for $x\to \pm \infty$, and a local maximum at $-3$ cannot occur without another local minimum at smaller $x$.

If $a<0$, $f$ approaches $-\infty$ for $x\to \pm \infty$, and a local minimum at $1$ cannot occur without another local maximum at larger $x$.

So a solution requires $a=0$, and there are effectively only 4 parameters left to determine.