# Find a,b, c and d in this function when given the inflection point and a local minimum point?

## The function is f(x)=$a {x}^{3} + b {x}^{2} + c x + d$ The local minimum is (3,3) and the inflection point is (2,5) I really appreciate some help :)

Apr 24, 2018

$f \left(x\right) = a {x}^{3} + b {x}^{2} + c x + d$

In this function there are $4$ unknowns , so $4$ equations are required to solve for $a , b , c , d$

#### Explanation:

$f \left(x\right) = a {x}^{3} + b {x}^{2} + c x + d$

$\Rightarrow f ' \left(x\right) = 3 a {x}^{2} + 2 b x + c$

And ;

$f ' ' \left(x\right) = 6 a x + 2 b$

At Local mimima $f ' \left(x\right) = 0$ :-

$\Rightarrow 3 a {x}^{2} + 2 b x + c = 0$ ...........where $x = 3$

$\Rightarrow 27 a + 6 b + c = 0$..........................................................$\left(1\right)$

Also $f \left(3\right) = 3$ :-

$\Rightarrow 3 = 27 a + 9 b + 3 c + d$...............................................$\left(2\right)$

And $f \left(2\right) = 5$ :-

$\Rightarrow 5 = 8 a + 4 b + 2 c + d$..................................................$\left(3\right)$

At Inflection point $f ' ' \left(x\right) = 0$ :-

$\Rightarrow 6 a x + 2 b = 0$....................where $x = 2$

$\Rightarrow 12 a + 2 b = 0$

$\Rightarrow 6 a + b = 0$........................................................................$\left(4\right)$

On Solving equations $\left(1\right) , \left(2\right) , \left(3\right) , \left(4\right)$ We finally get :-

$a = 1$

$b = - 6$

$c = 9$

$d = 3$

Thus the given function is :

$f \left(x\right) = {x}^{3} - 6 {x}^{2} + 9 x + 3$