Question about finding critical points

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jerry_v

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In my homework I am presented with the function y=x^4+4x and to find its critial points and local min/max. However I am having trouble finding the critical points and would appreciated any pointers/solutions. Thanks :)
 
Harry_the_cat said:
What do you understand by "critical points"?
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vaguely, like its the x intercepts of the derivatives. but i am unable to get it because i get to a point where i cube root a negative( which to my knowledge is not possible)
 
Please show us your work so we can where you are making any mistakes.
 
jerry_v said:
vaguely, like its the x intercepts of the derivatives. but i am unable to get it because i get to a point where i cube root a negative( which to my knowledge is not possible)
Click to expand...
That's your problem! With mathematics you can't be "vague". Definitions in mathematics (and the sciences in general) are "working definitions"- you use the precise words in problems and calculations.

The critical points, of a function, f, are points where either the derivative, df/dx, is equal to 0 or where the derivative does not exist. Here the function is \(\displaystyle f(x)= x^4+ 4x\). It's derivative is \(\displaystyle df/dx= 4x^3+ 4\).

That is a polynomial so exists everywhere. We are looking for points where \(\displaystyle df/dx= 4x^3+ 4= 0\). \(\displaystyle 4x^3= -4\). \(\displaystyle x^3= -1\). The only real number solution is \(\displaystyle x= -1\). The only '"critical point" is (-1, 0). The second derivative is \(\displaystyle d^2f/dx^2= 12x^2\) which is positive at x= -1. By the "second derivative test" (-1, 0) is a "local minimum".
 
..and a common way to recognize that this might be a mistake is getting 0 for the y-value
 
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