Easy way to find maximum/minimum value of polynomial

Question

I am given the following polynomial $$P(x) = (x-x_i)(x-x_{i+1})(x-x_{i+2})(x-x_{i+3})$$ We further know that $x_j-x_{j-1} = h$ . We need to determine the maximum value of function in the interval $(x_{i+2},x_{i+3})$. Is there an easy way to go about it, without much algebra? I have already tried substituting $(x-x_i)$ as $t$, but all I get is a cubic, whose roots are not trivial. (I get $x = x_i + \frac{3+\sqrt5}{2}h$, which is not easy to resubstitute. But, I know the answer is $h^4$.

Answer 1

Hint: An affine change of coordinates $t=x-x_i+3d$, with $d=\frac12h$, transforms $P(x)$ to $Q(t)=(t-3d)(t-d)(t+d)(t+3d)$.

Answer 2

Note that $$ P(t-\tfrac12(x_{i+1}+x_{i+2}))=(t-\tfrac32h)(t-\tfrac12h)(t+\tfrac12h)(t+\tfrac32h)=(t^2-\tfrac14h^2)(t^2-\tfrac94h^2).$$ Now substitute $u=t^2$, and note that the extreme value between $u=\tfrac14h^2$ and $u=\tfrac94h^2$ is achieved at the midpoint between those two zeros, at $u=\tfrac54h^2$. Compute the corresponding value as $$(\tfrac54h^2-\tfrac14h^2)(\tfrac54h^2-\tfrac94h^2)=-h^4.$$