Question: Consider the cubic function :
$y=\frac{x^3}{3} + x^2 + d\cdot x + 10$
where d is a constant, real number. If its stationary points are exactly square root 1332 units apart, find the value of d.
I found the derivative, keeping d in the equation, but when using the distance formula to create an equation for simultaneous equations I was lost. Can anyone help? The question, in theory, should only require knowledge of differential calculus.
For the stationary points it's first derivative is zero.
So, $$x^2+2x+d=0$$ $$\implies x=\sqrt{1-d}-1~or,~-\sqrt{1-d}-1$$ When $x=\sqrt{1-d}-1,y=\dfrac{1}{3}\big(2d\sqrt{1-d}-3d-2\sqrt{1-d}+32)$
And when $x=-\sqrt{1-d}-1,y=\dfrac{1}{3}\big(-2d\sqrt{1-d}-3d+2\sqrt{1-d}+32\big)$
Now if you take the distance between these two points and after equalling it with $\sqrt{1332}$,after some simplification you will find- $$4(1-d)+\dfrac{16}{9}(1-d)^3=1332$$ $$Let, 1-d=p$$ $$\implies \dfrac{16}{9}p^3+4p-1332=0$$ $$p=9[only~real~root]$$ $$Hence,~~d=-8$$