How to minimize the maxinum of the absolute value of the following cubic polynomial on the interval [-1,1] with two parameters (b,d)?
$$ -\frac{5{x}^3}{2}+720d{x}^2 + {x}^2 +720b \left( x +1 \right)+5x+2 $$
I think there should be a method to calculate an analytical expression of b and d. But this problem is a little tricky for me. I would also be grateful if anyone can provide some books or articles helping solving such problems.
Let
$$f_{b,d}(x):=-\frac{5{x}^3}{2}+(720d+1)x^2 + 720b \left(x+1 \right)+5x+2$$
with
$$f'_{b,d}(x)=-\frac{15{x}^2}{2}+2(720d+1)x+(720b+5)$$
which is $0$ (corresponding to a maximum of $f_{a,b}$) if
$$x_1=\frac{2}{15}((720d+1)-\sqrt{\Delta}) \ \text{and} \ x_2=\frac{2}{15}((720d+1)+\sqrt{\Delta})$$
with $\Delta=(720d+1)^2+\frac{15}{2}(720b+5)$
Then you have different cases.
For example, if neither $x_1$, nor $x_2$ is within $[-1,1]$, you just have to get the values at the boundaries: the maximum can therefore be defined as :
$$M_{a,b}=max( |f_{b,d}(-1)|,|f_{b,d}(1)|) $$
Then, only, in a second step, you will minimize this expression.
I advise you to use Desmos. I have "played" with different values of $b,d$ : it looks that it is with $b=0,d=0$ that the minimum is obtained.