#### To determine

**To find:**

The asymptotes and inflection points of y=ln(x2+c) and describe the graph of the function for different values of *c*

#### Answer

f(−c)=ln(c+c)=ln2cf(c)=ln(c+c)=ln2c

#### Explanation

**Inflection Points**:

Inflection points exist for only positive values of c, can be found by differentiation

y=ln(x2+c)y′=2xx2+cy″=(x2+c).2−2x.2x(x2+c)2=2(c−x2)(x2+c)2

y″=0⇒c−x2=0⇒x=±c

**Asymptotes:** The asymptotes of the function y=ln(x2+c) can be found for different values of c

If c>0, then y=ln(x2+c)>0, so no vertical asymptote is possible ( Domain is ℝ)

If c≤0, then x2+c>0⇒x=±−c are the two asymptotes.

Therefore

f(−c)=ln(c+c)=ln2cf(c)=ln(c+c)=ln2c

**Conclusion:**

Inflection points exist for only positive values of c, and are given by

(−c,ln2c) and (c,ln2c)