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

Question

William Abdalla · Accepted Answer

Look for zeros on the first and second derivatives.

Start by finding the first and second derivative. #f(x)=1/x-3x^3=x^(-1)-3x^3# rewrite #f(x)# such that it fits the power rule. #f'(x)=-x^(-2)-9x^(2)=-1/(x^(2))-9x^2# Now look at the first derivative: both #-1/x^2# and #- 9x^2# are negative for real values of x. Additionally, the first derivative is not defined for #x=0# and demonstrates asymptotic behavior for that value. Thus #f(x)# shall consist of two branches and given that #f'(x)<0#, #f(x)# is decreasing over both branches. Zeros and signs on the second derivative give additional information about the concavity of and inflection points (if any) on #f(x)#. #f''(x)=2x^(-3)-18x=2/(x^3)-18x# Solving #f''(x)=0# gives #x^4=1/9#, #x=pm 1/sqrt(3)# #f''(x)# shows asymptotic behavior at #x=0# so checking the value of #f''(x)# on the four open intervals shall give a comprehensive conclusion on the sign of #f''(x)# over the whole real domain. Conclusion: taking information from the second derivative into account: #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 #f(x)# graph{1/x-3x^3 [-1.5, 1.5, -5, 5]}

Paisley Johnson · Answer

undefined To graph $ f(x) = \frac{1}{x} - 3x^3 $ using information from the first derivative, follow these steps:

1. Find the critical points by setting the first derivative equal to zero and solving for $ x $.
2. Determine the intervals where the first derivative is positive and negative.
3. Use the behavior of the first derivative to determine the increasing and decreasing intervals.
4. Analyze the concavity of the function using the second derivative.
5. Use the critical points, increasing/decreasing intervals, and concavity to sketch the graph of the function.