Graph y=3x^3-36x-2

Problem

y=3*x3−36*x−2

Solution

1. Identify the function type and determine the end behavior. Since this is a cubic polynomial with a positive leading coefficient, the graph falls to the left and rises to the right.

x→∞,y→∞

x→−∞,y→−∞

2. Find the y-intercept by setting x=0

y=3*(0)3−36*(0)−2

y=−2

3. Find the first derivative to locate critical points where the slope is zero.

d(y)/d(x)=9*x2−36

4. Solve for critical points by setting the first derivative to zero.

9*x2−36=0

x2=4

x=±2

5. Determine the coordinates of local extrema by substituting the critical points back into the original equation.

y(2)=3*(2)3−36*(2)−2=24−72−2=−50

y*(−2)=3*(−2)3−36*(−2)−2=−24+72−2=46

The local maximum is at (−2,46) and the local minimum is at (2,−50)

6. Find the second derivative to determine concavity and the inflection point.

d2(y)/(d(x)2)=18*x

7. Locate the inflection point by setting the second derivative to zero.

18*x=0

x=0

The inflection point is at (0,−2) where the graph changes from concave down to concave up.

Final Answer

y=3*x3−36*x−2

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