Solving math problems involving extra variable (p)? I have a very hard time solving problems in which you have to solve for the additional unknown variable. I would like to know whether there is some method I can learn or approach I can simulate in order to solve these kinds of problem. Any help is highly appreciated!
Some examples of what kind of problems I'm talking about:
First of all we have another question I asked. Take a look: 


*

*Calculate unknown variable when surface area is given. (Calculus)
Then we have some problems I will describe below:
First problem:
For every value of p the following function is given: $$f_{p}(x)=x^3-3px^2-9x$$
Show with an exact calculation why $\,\,\,\, f_{p}(x) \,\,\,\,$ has two extreme values for every value of p.
Second problem: 
Given are the functions: $$f_{p}(x) = x^3 +5x^2+px$$
Investigate with the help of an algebraic calculation for which values of p the graphs of $\,\,f_{p}(x)\,\,$ and the parabola $\,\,y=x^2+2x\,\,$ touch each other.
Third problem:
For every value of p the following function is given $\,\,f_{p}(x)=x^4-px^3\,\,$
a: Proof that the graphs of $\,f_{p}\,$ have for every value of p  2 inflection points, except for $\,p=0\,$.
b: Give an exact calculation from which is evident for which values of p the slope of the graph $\,f_{p}\,$ in one of its inflection points equals 2. 

I have a tremendously hard time figuring out how to solve these problems. Now, when I've seen these problems before I can solve them but not the first time.. Would you please make the problems above and describe your methods to me? What do you do?
-Bowser



*

*With exact I mean without use of a calculator.

*With calculate algebraically/algebraic calculation I mean without the use of an calculator, except at the end of the problem
 A: *

*Extrema are found by canceling the first derivative.


$$3x^2-6px-9=0$$
The discriminant $36(p^2+3)$ is always positive.


*The two graphs must have a common point, with equal derivatives.


$$x^3+5x^2+px=x^2+2x,\\3x^2+10x+p=2x+2.$$
Simplifying by $x$ and subtracting,
$$2x^2+4x=0$$ so that $x=0$ or $x=-2$, giving $p=2$ or $p=6$.


*a. Inflections occur at zeroes of the second derivative,


$$12x^2-6px=0$$ so that $x=0$ or $x=\frac p2$. When $p=0$, these two roots merge in a double one.
b. The slope is $4x^3-3p^2$, which should equal $2$ at $0$ or at $\frac p2$.
So either $-3p^2=2$, which is impossible, or $\frac{p^3}2-3p^2=2$, which you haven't learnt how to solve.
A: "Problems with a parameter"   turn up everyday in all sorts of applications. It is therefore of utmost importance that you understand the logic behind them.
You are not given one situation $\Psi$ to analyze, like finding the maximum of some function, or the points of intersection between a line and an ellipse. Rather you are given an infinite family $(\Psi_p)_{p\in{\mathbb R}}$ of "presumably similar" situations. These situations are "numbered" by the parameter variable $p$. This means that with not much extra effort from his side the teacher has confronted you with an infinity of problems instead of just one.
The following circumstance makes such compound problems particularly interesting: While all situations $\Psi_p$ are described by  formulas looking alike, the geometric picture can look radically different for different values of $p$. Often there is a "catastrophic" value $p_*$ where the picture changes abruptly. An example: For $p<p_*$ the $\Psi_p$ are ellipses, and for $p>p_*$ the $\Psi_p$ are hyperbolas.
The three compound problems in your question are all of the same type: You are given a family $(f_p)_{p\in{\mathbb R}}$ of functions $x\mapsto f_p(x)$, and are asked, for which $p$ the graph of $f_p$ has a certain property. They all can be dealt with in the same way: 
Assume for the moment that $p$ is fixed, i.e. has a given value  known to you (or  hidden in a box). Now analyze $x\mapsto f_p(x)$ as a function of $x$ alone, according to the rules you have learnt: Find the maximum, inflection points, points of intersection with some other given graph, etc. Your findings will depend on the momentary value of $p$: The graph of $f_p$ might have inflection points for certain values of $p$ and none for others, the maximum might lie in the interior of the considered $x$-interval $[a,b]$ for certain values of $p$, and might be the left endpoint for other values of $p$, etc. Report your findings in terms of a list sorted according to the $p$-domains in which essentially different phenomena are to be seen.
