the sign of the first derivative

Maybe this is a question that is a bit redundant.

I just started the chapter on curve sketching. The first section is about local maxima and minima.

How do I determine the sign of the first derivative to see if it changes at the possible turning point? I know that a function like f(x) = x^3 doesn't have a turning point.

Do I need to start sketching the curve by plotting the possible turning points or is there a more intuitive way to determine if there is a sign change.

thanks

Re: the sign of the first derivative

The derivative of a function has the intermediate value property. Find all values of x where the derivative is zero or where it does not exist. Suppose you find the values $\displaystyle \{a_1,\ldots,a_n\}$ where the derivative is zero or it does not exist. Then consider the intervals: $\displaystyle (-\infty,a_1), (a_1,a_2), \ldots, (a_{n-1},a_n), (a_n,\infty)$. For each interval, if the derivative is positive, it is positive for every value of the interval. If it is negative, it is negative for every value of the interval. So, just pick any point in the interval and plug it into the derivative. That will tell you the sign of the derivative on the entire interval.

Re: the sign of the first derivative