Find b and d in cubic equation

Question

Find the $b$ and $d$ in equation: $$ y= -x^3 + bx^2 + 4x + d $$ The x-intercept is $(2,0)$ and it is point of inflection, but I don't know how to apply it to help solve the problem (point 2,0 is only given point). I get to d= -4b but I got stuck...

Answer 1

It is given to you that $(2,0)$ is a point of infliction. To use that information, differentiate the function twice

$$f(x) = -x^3+bx^2+4x-d$$

$$f'(x) = -3x^2+2bx+4$$

$$f''(x) = -6x+2b$$

At the point of infliction of a cubic, $f''(x)$ will be $0$.

$$-6(2)+2b=0$$

$$b=6$$

In the question you already have a relation between $b$ and $d$, use it to find $d$.