Question

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

Answer 1

Let the polynomial be $P\left(x\right)=a{x}^2+b{x}^2+cx+d$

According to given conditions
$P\left(1\right)=a+bc+d=10$
$P\left(1\right)=a+b+c+d=6$
Also $P\left(1\right)=3a2b+c=0$
and $P\left(1\right)=6a+2b=0$

$\Rightarrow 3a+b=0$

Solving for $a,b,c,d$ we get $P\left(x\right)=x^3-3x^3-9x+5\Rightarrow P^{{}^{\prime }}\left(x\right)=3x^2-6x-9=3(x+1)(x-3)$

$\Rightarrow x=-1$ is the point of maximum and $x=3$ is the point of minimum.
Hence distance between $(-1,10)$ and $(3,-22)$ is $4\sqrt{65}$ units.