Thread: Increasing Function

Find all exact values of for which is always an increasing function.

My working:

First notice the domain:

We require all for all x within this domain.

Thus

Here we place a restriction on the original domain for [since , because derivative is not defined at this point]

Thus for all

Thus we now need to ensure

Solving for x = 0 yields:



Notice ie for all a

Now what?

Originally Posted by usagi_killer

Find all exact values of for which is always an increasing function.

My working:

First notice the domain:

We require all for all x within this domain.

Thus

Here we place a restriction on the original domain for [since , because derivative is not defined at this point]

Thus for all

Thus we now need to ensure

Solving for x = 0 yields:



Notice ie for all a

Now what?

Those two points divide the number line into three intervals in each of which the derivative has the same sign. Calculate the derivative at one point in each interval to determine whether that sign is "+" or "-".