Graph f(x)=x^3-3x

Problem

ƒ(x)=x^3−3*x

Solution

1. Find the x-intercepts by setting ƒ(x)=0 and factoring the expression.

x^3−3*x=0

x*(x^2−3)=0

x*(x−√(,3))*(x+√(,3))=0

The intercepts are (0,0) (√(,3),0) and (−√(,3),0)

2. Find the y-intercept by evaluating ƒ(0)

ƒ(0)=0^3−3*(0)=0

The intercept is (0,0)

3. Determine the symmetry of the function by checking if it is even or odd.

ƒ*(−x)=(−x)^3−3*(−x)=−x^3+3*x=−(x^3−3*x)=−ƒ(x)

The function is odd, meaning it is symmetric about the origin.

4. Find the critical points by taking the first derivative and setting it to zero.

(d(x^3)−3*x)/d(x)=3*x^2−3

3*x^2−3=0

x^2=1

x=±1

5. Identify local extrema by evaluating the function at the critical points.

ƒ(1)=1^3−3*(1)=−2

ƒ*(−1)=(−1)^3−3*(−1)=2

The local maximum is at (−1,2) and the local minimum is at (1,−2)

6. Find the inflection point by taking the second derivative and setting it to zero.

(d(3)*x^2−3)/d(x)=6*x

6*x=0

x=0

The inflection point is at (0,0) where the concavity changes from downward to upward.

7. Determine end behavior by looking at the leading term x^3
   As x→∞ ƒ(x)→∞
   As x→−∞ ƒ(x)→−∞

Final Answer

To graph ƒ(x)=x^3−3*x plot the intercepts at (0,0) and (±√(,3),0) the local maximum at (−1,2) the local minimum at (1,−2) and the inflection point at (0,0) then connect them following the cubic end behavior.

Want more problems? Check here!