# What are the absolute extrema of  f(x)= |sin(x) + ln(x)| on the interval (0 ,9]?

May 21, 2017

No maximum. Minimum is $0$.

#### Explanation:

No maximum
As $x \rightarrow 0$, $\sin x \rightarrow 0$ and $\ln x \rightarrow - \infty$, so

${\lim}_{x \rightarrow 0} \left\mid \sin x + \ln x \right\mid = \infty$

So there is no maximum.

No minimum

Let $g \left(x\right) = \sin x + \ln x$ and note that $g$ is continuous on $\left[a , b\right]$ for any positive $a$ and $b$.

$g \left(1\right) = \sin 1 > 0$ $\text{ }$ and $\text{ }$ $g \left({e}^{-} 2\right) = \sin \left({e}^{-} 2\right) - 2 < 0$.

$g$ is continuous on $\left[{e}^{-} 2 , 1\right]$ which is a subset of $\left(0 , 9\right]$.

By the intermediate value theorem, $g$ has a zero in $\left[{e}^{-} 2 , 1\right]$ which is a subset of $\left(0 , 9\right]$.

The same number is a zero for $f \left(x\right) = \left\mid \sin x + \ln x \right\mid$ (which must be non-negative fo all $x$ in the domain.)