How to show that a derivative of a function is always increasing or decreasing? (1 Viewer)

[xXnukerrrXx]

Im kinda stuck on proving that the derivative of a function is always increasing or decreasing e.g. 2x^3-3x^2+5x+1 and [always increasing], -x^3+2x^2-5x+7 [always decreasing]. What is the proper way of concluding why its always positive/negative

 

[Drongoski]

A function of x, f(x), is always increasing if its derivative f'(x), is always positive and always decreasing if f'(x) is always negative.

e.g.





Here f'(x) is ALWAYS positive - so it is a monotone increasing function of x.

 

[mitchy_boy]

Show that the derivative is always greater than zero? Show that it's minimum point is positive, and thus that it's always positive.
The second funtion both increases and decreases so I dunno, wtf's going on there.

 

[funnytomato]

Im kinda stuck on proving that the derivative of a function is always increasing or decreasing e.g. 2x^3-3x^2+5x+1 and [always increasing], -x^3+2x^2-5x+7 [always decreasing]. What is the proper way of concluding why its always positive/negative

what drongoski did

and for the 2nd one, you should be able to factorise f'(x) as -3* (x-2/3)^2 - 11/3 <= - 11/3 < 0, as -3* (x-2/3)^2 <= 0
which means the derivative is always negative, hence the function is always decreasing