Question

Analyzing a Graph In Exercise, analyze and sketch the graph of the function. Lable any relative extrema, points of inflection, and asymptotes. See Example 6.
y = x - ln x

Answer 1

Answer:

x=1 is the minima

No point of inflection

y axis is asymptote

Step-by-step explanation:

A function is given by

$y=x-lnx$

Let us use derivatives to find extrema point of intersection etc

$y' = 1-1/x\\y"=1/x^2\\y"' = 2/x^3$

We find that second derivative cannot be equal to 0 for any value of x.

Hence there are no points of inflection.

Equate I derivative to 0

x =1

At x=1 I derivative is 0 and II derivative positive. So minima at x=1 and y =1

ln x is not defined for 0 or negative values of x

Hence asymptote is x=0 or y axis

Domain is (0,infty)

Range is (1,infty)

Since I derivative is negative for x <1 and positive for x>1

decreasing in (0,1) and increasing in (1,infty)