# $P(x)$ is a polynomial of least degree with a local maximum of $6$ at $x=1$ and minimum of $2$ at $x=3$. Find $P^\prime(0)$.

Let $$P(x)$$ be a real polynomial of least degree which has a local maximum at $$x=1$$ and minimum at $$x=3$$.If $$P(1)=6$$ and $$P(3)=2$$, then find $$P'(0)$$.

Method: Since the smallest polynomial whose derivative gives 2 root would be a cubic equation. So I assumed my function to be

$$P(x) = ax^3+ bx^2+cx+d$$

Solving the condition above gives me $$b=-6a$$ and $$c=11a$$. I am stuck after this. Can anyone tell me how to proceed from here?

• Well, you also know that $P(1)=6,P(3)=2$. – lulu Sep 23 '18 at 14:02
• Note: I don't agree that $c=11a$. You have $P'(x)=3ax^2+2bx+c=\lambda(x-3)(x-1)=\lambda x^2-4\lambda x+3\lambda\implies a=3\lambda,\,2b=-4\lambda=-12a,\,c=3\lambda=9a$. No? – lulu Sep 23 '18 at 14:06

Hint: Consider the System $$a+b+c+d=6$$ $$27a+9b+3c+d=2$$ $$3a+2b+c=0$$ $$27a+6b+c=0$$
You attempt was right to assume a function of the type $$P(x)=ax^3+bx^2+cx+d$$ as the solution. Rewriting the conditions as $$P(1)=6, P(3)=2, P'(1)=0$$ and $$P'(3)=0$$ leads to a system of four equations with four variables $$a,b,c$$ and $$d$$.
If you use lulu's idea, then you should get the answer quicker because you will be solving a simultaneous system in two variables, instead of four variables. First, if $$P(x)$$ is of lowest degree, then we expect $$P(x)$$ to be a cubic polynomial. Since $$P'(x)$$ is a quadratic polynomial with two roots $$x=1$$ and $$x=3$$, we have $$P'(x)=3a(x-1)(x-3)=3ax^2-12ax+9a$$ for some constant $$a$$ (the factor $$3$$ is there simply to make the integration easier). Thus, $$P(x)=ax^3-6ax^2+9ax+c$$ for some constant $$c$$. Since $$P(1)=6$$ and $$P(3)=2$$, we get $$4a+c=6\text{ and }c=2\,,$$ so $$a=1$$ and $$c=2$$. That is, $$P(x)=x^3-6x^2+9x+2\,,$$ with $$P'(x)=3(x-1)(x-3)=3x^2-12x+9\text{ and }P''(x)=6(x-2)\,.$$ This makes $$P''(1)=-6<0$$ and $$P''(3)=6>0$$, so $$x=1$$ and $$x=3$$ are the local maximum and the local minimum, respectively. Hence, $$P(x)$$ is indeed the desired polynomial.