# How do you graph #f(x)=1/x-3x^3# using the information given by the first derivative?

##### 1 Answer

Look for zeros on the first and second derivatives.

#### Explanation:

Start by finding the first and second derivative.

rewrite

Now look at the first derivative: both

Zeros and signs on the second derivative give additional information about the concavity of and inflection points (if any) on

Solving

#(-infty, -1/sqrt(3))# #(-1/sqrt(3), 0)# #(0, 1/sqrt(3))# , and#(1/sqrt(3), infty)#

shall give a comprehensive conclusion on the sign of

#f''(x)>0# for#x in (-infty, -1/sqrt(3))# and#x in (0, 1/sqrt(3))# ;#f''(x)<0# for#x in (-1/sqrt(3),0)# and#x in (1/sqrt(3),0)# .

Conclusion:

#f(x)# demonstrates asymptotic behavior on#x=0# #f(x)# is decreasing on both of its branches (from 1st derivative).

taking information from the second derivative into account:

#f(x)# is concave upward on#x in (-infty, -1/sqrt(3))# and#x in (0, 1/sqrt(3))# -
#f(x)# is concave downward on#x in (-1/sqrt(3),0)# and#x in (1/sqrt(3),0)# . -
#f(x)# has inflection points at#x=pm 1/sqrt(3)#

FYI Here's a plot of