How to find the critical points of a polynomial?

Question

I need help finding the critical points of the function $x^5+x^4-2x^2$, I don't understand, can someone help show me how to find the critical points please

Answer 1

Hint: Try differentiating $f(x) = x^5 + x^4 - 2x^2$.

Then try to find the values of $x$ at which $f^\prime(x) = 0.$ One such value for $x$ will be clear. Then try to find the second root, by approximating, if necessary.

See for example, the plot of your function close to the origin:

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In general, for finding critical points $c$ of a function $f$ (not necessarily a polynomial), the following may help:

Critical point checklist:

Given a function $f(x)$ defined on an interval $J$ and a point $c$ in $J$ in its domain, then...

either f(x) is continuous at x = c or it is not. If f(x) is not continuous at x = c, then c is a critical point. If f(x) IS continuous at x = c, then

    either c is an endpoint of J or it is not. If c is an endpoint of J, then c is a critical point. If c is NOT an endpoint of J, then

        either f '(c) is defined or it is not. If f '(c) is not defined, then c is a critical point. If f '(c) IS defined, then

            either f '(c) = 0 or f '(c) 0. If f '(c) = 0, then c is a critical point. If f '(c) 0, then f '(c) is not a critical point. 

Answer 2

You take the derivative-what do you get? Then set it to zero. You should have a fourth degree derivative. One root is obvious. There is a second real one that you can only approximate unless you want to solve a messy cubic.