Role of differentiation in a polynomial

Question

I have learnt that, to find maximum or minimum value of a polynomial $p(x)$, we take its derivative, equate it to zero, solve for x, find the maxima and minima, and then put the value of x in the original equation. But when we take the derivative of $p(x)$, what exactly are we doing? Also, what do we do if we are unable to solve for x even after taking the derivative?

Answer 1

When I take the derivative of a function $p(x)$, at least in the back of my mind, I am finding another function $p'(x)$ with the property that $$p'(x)=\lim_{h\rightarrow0}\frac{p(x+h)-p(x)}{h}.$$ Essentially, I am finding another function that contains the instantaneous rate of change of the function $p(x)$ at various values for the independent variable $x$.

When I think of the zeroes of the derivative corresponding to the extrema (max/min) of the function, I think about how the extrema correspond to the points at which the instantaneous rate of change (or the derivative) on the original function $p(x)$ is identically zero, which is to say that the slope of the tangent line at that point is $0$.

Some derivatives are not defined for all $x$. This just means the derivative does not exist at those values. Other derivatives are defined at various values of $x$, but the task of actually finding them by pure algebraic methods is quite beyond any human's reach. We use approximation methods to find these when necessary.

Answer 2

We are finding the critical point of the function because at this critical point the rate of change of said function f is 0. Think of a person on top of a hill, here I am talking about a function with a maxima. Now think of a function with a minima or another analogy would be a person in a ditch. When the rate of change is no longer monic, we've reached an extrema (assuming the function is differentiable there). If we aren't able to solve for x one of two things may be the case, we aren't cleaver enough or the function has no extrema, i.e on no interval is the function bounded or equivalently on no interval is there a local max or min.

Hope this helps!

Answer 3

A polynomial is not bounded in $R$. In case you want to find a maximum or a minimum of a polynomial in $R$, you can find at least one of them. The value that you get from derivative is just a local maximum or minimum.