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.

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- Mar 27th 2008, 04:20 PMmathleteFamily 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. - Mar 27th 2008, 04:31 PMTheEmptySet
from the given info we know that the function looks like

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

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

max and min points occur when the dervitive is zero so

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

Solving this system for a and b gives us the polynomial

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

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

I hope this helps

Good luck