Distance between local maximum and local minimum

Question

If $P (x)$ be a polynomial of degree $3$ satisfying $P (-1)=10,P (1)=-6$ and $P(x)$ has maximum at $x=-1$ and $P'(x)$ has minima at $x=1$.Find the distance between the local maximum and local minimum of the curve.

Answer 1

Let the polynomial be $P(X)= ax^3+bx^2+cx+d$.

As per the given conditions we have :

$P(-1) = -a+b-c+d=10$

$P(1) = a+b+c+d=-6$

Also, $P'(-1)=3a-2b+c=0$

And $P"(1) = 6a+2b=0$

Solving for a,b,c,d gives

$P(X)=x^3-3x^2-9x+5$

=> $P'(X)=3x^2-6x-9=3(x+1)(x-3)$

This implies $x=-1$ is the point of maximum and $x=3$ is the point of minimum

Hence, the distance between $(-1,10)$ and $(3,-22)$ is $4\sqrt{65}$