Question

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

Answer 1

No maximum. Minimum is `0`.

Explanation:

No maximum
As `xrarr0`, `sinxrarr0` and `lnxrarr-oo`, so

`lim_(xrarr0) abs(sinx+lnx) = oo`

So there is no maximum.

No minimum

Let `g(x) = sinx+lnx` and note that `g` is continuous on `[a,b]` for any positive `a` and `b`.

`g(1) = sin1 > 0` `" "` and `" "` `g(e^-2) = sin(e^-2) -2 < 0`.

`g` is continuous on `[e^-2,1]` which is a subset of `(0,9]`.

By the intermediate value theorem, `g` has a zero in `[e^-2,1]` which is a subset of `(0,9]`.

The same number is a zero for `f(x) = abs(sinx+lnx)` (which must be non-negative fo all `x` in the domain.)