[Math] How to know this critical point is the local maximum

calculus

Let
$f(x) = 4ln(10x)-4x$, $x>0$

I found the derivative and the critical point was 1, however, how do I know whether or not this is the local maxima or minima ?

The next part of the question also asks:

1) a.The interval on the left of the critical point is ….

b.On this interval, $f$ is….. while $f'$ is …..

2) a.The interval on the right of the critical point is ….

b.On this interval, $f$ is….. while $f'$ is …..

Any explanation on how to figure our 1 and 2 , along with how to determine whether the critical point is the maxima or minima would be greatly appreciated!

Thanks!

Best Answer

I would suggest taking values to the left and right of $x$ and substituting it into $f(x)$. For example, if we have $f'(1)=0$, then take values such as 0.99, 0.98, and 1.01, 1.02, and investigate the "behaviour" of the graph at those points.

Alternatively, you can take use the "Second Derivative Test", which gives you a good idea of the behaviour of $f(x)$. Typically,

  1. when $f''(x)<0$, the point is a local maximum, and,
  2. when $f''(x)>0$, the point is a local minimum.

However, this is not always the case (for example, $x=0$ for $y=x^3$), so for a more foolproof method, "probe" the behaviour near the point $x=1$. You will find that it is a maximum point.

Hope this helps!

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