Question

Analyzing a Graph In Exercise, analyze and sketch the graph of the function. Label any relative extrema, points of inflection, and asymptotes.

y = In x^2/ 9 - x^2

Answer 1

analyzed & sketched

Explanation:

We are given the function:

`y = (\ln^2(x))/(9-x^2)`

The first derivative to find the region for increase and decrease:

`y'= ((2\ln(x))/(x) (9-x^2)+2x\ln^2(x))/((9-x^2)^2) =(2\ln(x)(9-x^2+x^2\ln(x)))/(x(9-x^2)^2) =0`

decreasing in (0,1), increasing in (1,3)∪(3,∞)

So, x = 1 is local minimum.

x = 3 is vertical asymptote.

The second derivative to find concave up and concave down:

`y''=(2\left(\left(x^2-9\right)^2+3x^2\left(x^2+3\right)\ln^2\left(x\right)+\left(-5x^4+54x^2-81\right)\ln\left(x\right)\right))/(x^2\left(9-x^2\right)^3)=0`

concave up in (0,3), concave down in (3,∞)

The sketch is given in the attachment.