# Math Help - Family of Functions!!

1. ## Family of Functions!!

Find a formula for the following function.
A cubic polynomial with a local maximum at x = 7, a local minimum at x = 9, a y-intercept of 4, and an x^3 term whose coefficient is 1.

2. Originally Posted by mathlete
Find a formula for the following function.
A cubic polynomial with a local maximum at x = 7, a local minimum at x = 9, a y-intercept of 4, and an x^3 term whose coefficient is 1.
from the given info we know that the function looks like

$f(x)=x^3+ax^2+bx+4$ so taking the derivative we get

$f'(x)=3x^2+2ax+b$

max and min points occur when the dervitive is zero so

$f'(7)=0=147+14a+b$ and $f'(9)=0=243+18a+b$

Solving this system for a and b gives us the polynomial

we get $a=-24 \mbox{ and } b=189$

So we get $f(x)=x^3-24x^2+189x+4$

I hope this helps

Good luck