Science:Math Exam Resources/Courses/MATH110/April 2013/Question 02 (a)/Solution 1

A simple-like parabola with a zero at three will work. The other zero one can be anywhere, for no particular reason lets make it at 5. So the function

$\displaystyle f(x)=(x-3)(x-5)=x^{2}-8x+15$

satisfies all three conditions.

To double check,

${\begin{aligned}f(3)&=(3-3)(3-5)=0\\f'(3)&=2(3)-8=-2<0\\f''(3)&=2>0\\\end{aligned}}$

Let's see the picture

Note: Many other functions are possible as well, it does not even have to be a second order polynomial.