Graph y=x-x^3

Problem

y=x−x3

Solution

1. Find the intercepts by setting y=0 to find the x-intercepts and x=0 to find the y-intercept.

0=x*(1−x2)

0=x*(1−x)*(1+x)

The x-intercepts are (0,0) (1,0) and (−1,0) The y-intercept is (0,0)

2. Determine symmetry by checking if the function is even or odd.

ƒ*(−x)=(−x)−(−x)3

ƒ*(−x)=−x+x3

ƒ*(−x)=−(x−x3)

Since ƒ*(−x)=−ƒ(x) the function is odd and symmetric about the origin.

3. Find the first derivative to identify critical points and intervals of increase or decrease.

d(y)/d(x)=1−3*x2

Setting the derivative to zero:

1−3*x2=0

x2=1/3

x=±1/√(,3)

4. Find the second derivative to determine concavity and inflection points.

d2(y)/(d(x)2)=−6*x

Setting the second derivative to zero:

−6*x=0

x=0

The graph is concave up for x<0 and concave down for x>0 with an inflection point at (0,0)

5. Analyze end behavior by looking at the leading term −x3
   As x→∞ y→−∞
   As x→−∞ y→∞

6. Sketch the graph using the points and properties found. The curve starts from the upper left, passes through (−1,0) reaches a local minimum at x=−1/√(,3) passes through the origin (inflection point), reaches a local maximum at x=1/√(,3) passes through (1,0) and continues to the lower right.

Final Answer

y=x−x3

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