I'm stuck with this kind of math problems when you have to find a function with given local extrema points or inflection points. Are there any general formula or method to find the function?
For instance, find a function with a local maximimum point at $(1, 2)$ and a local minimum point at $(5, -4)$?
How would you solve it (with derivative)?
I start with this:
$$f(1) = 2 \qquad f(5) = -4 \qquad f'(1) = 0 \qquad f'(5) = 0\\
f''(1) < 0 (\text{max point}) \\
f''(5) > 0 (\text{min point})$$
and then try to solve $$f(x) = ax^3 + bx^2 + cx + d$$
but it doesn't work.
Best Answer
$f(1)=2$ gives $a+b+c=d=2.$ And $f(5)=-4$ gives $125a+25b+5c+d=-4.$
Also $f'(1)=0$ gives $3a+2b+c=0,$ and $f'(5)=0$ gives $75a+10b+c=0.$
This is four linear equations in four unknowns. I just checked, and there is a unique solution.
And note with the values plugged in the function $f(x)$ has the required properties.