Graph f(x)=x^3-25x

Problem

ƒ(x)=x3−25*x

Solution

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

x3−25*x=0

x*(x2−25)=0

x*(x−5)*(x+5)=0

x=0,x=5,x=−5

2. Find the y-intercept by evaluating the function at x=0

ƒ(0)=0−25*(0)

ƒ(0)=0

3. Determine the end behavior by looking at the leading term x3

As *x→∞,ƒ(x)→∞

As *x→−∞,ƒ(x)→−∞

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

(d(x3)−25*x)/d(x)=3*x2−25

3*x2−25=0

x2=25/3

x=±5/√(,3)≈±2.89

5. Calculate the local extrema by plugging the critical points back into ƒ(x)

ƒ(2.89)≈−48.11

ƒ*(−2.89)≈48.11

6. Sketch the graph using the intercepts (−5,0),(0,0),(5,0) the local maximum at ≈(−2.89,48.11) and the local minimum at ≈(2.89,−48.11)

Final Answer

ƒ(x)=x*(x−5)*(x+5)

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