How do you graph #f(x)=1/3x^3-6x# using the information given by the first derivative?

Question

Aurora Alvarado · Accepted Answer

From #f'(x) = 0, f(x)# has turning points at #x=+-sqrt6#

#f(x)= 1/3x^3-6x#

#f'(x) = cancel3 x^2/cancel 3 -6#

#f(x)# has tuning points where #f'(x) = 0# I.e. when #x^2 = 6 -> x=+-sqrt6#

From the graph below, #f(-sqrt6)# has curvature concave down #-> # a local maximum and #f(+sqrt6)# has curvature concave up #-># a local minimum.

Also, #f(x) =0# when #x(x^2/3-6)=0# I.e. when #x=0# or #x= +-sqrt18 = +-3sqrt2#

graph{1/3x^3-6x [-25.67, 25.65, -12.83, 12.84]}

Charles Anctil · Answer

undefined To graph $ f(x) = \frac{1}{3}x^3 - 6x $ using information from the first derivative: 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 or negative to identify where the function is increasing or decreasing. 3. Use this information to sketch the graph of the function. First, find the first derivative: $$ f'(x) = x^2 - 6 $$ Set $ f'(x) $ equal to zero: $$ x^2 - 6 = 0 $$ Solve for $ x $: $$ x^2 = 6 $$ $$ x = \pm \sqrt{6} $$ These are the critical points. Next, test intervals around the critical points in $ f'(x) $ to determine the sign of the derivative: - For $ x < -\sqrt{6} $, choose $ x = -2 $, $ f'(-2) = (-2)^2 - 6 = -2 $, so $ f'(x) < 0 $. The function is decreasing in this interval. - For $ -\sqrt{6} < x < \sqrt{6} $, choose $ x = 0 $, $ f'(0) = -6 $, so $ f'(x) < 0 $. The function is decreasing in this interval. - For $ x > \sqrt{6} $, choose $ x = 2 $, $ f'(2) = 2^2 - 6 = 2 $, so $ f'(x) > 0 $. The function is increasing in this interval. Now, plot the critical points and sketch the graph accordingly: - The function decreases from $ -\infty $ to $ -\sqrt{6} $, has a local minimum at $ -\sqrt{6} $, decreases again from $ -\sqrt{6} $ to $ \sqrt{6} $, has a local maximum at $ \sqrt{6} $, and increases from $ \sqrt{6} $ to $ \infty $. - The behavior around the critical points helps sketch the shape of the graph.