Can we get a function in the following form and properties ?
$f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 )$ $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has two distinct real roots.
Can anyone help me to find a function in that way ?
By using calculus , we can say for sure $f'(x)$ will have at least three distinct real roots and $f''(x)$ has at least two distinct real roots.
But I am finding a function in that form such that $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has exactly $2$ distinct real roots.