Find the Local Maxima and Minima y=x^3-6x^2-135x

Problem

y=x3−6*x2−135*x

Solution

1. Find the first derivative of the function to determine the slope of the tangent line.

d(y)/d(x)=3*x2−12*x−135

2. Set the derivative to zero to find the critical points where the slope is horizontal.

3*x2−12*x−135=0

3. Solve for x by factoring the quadratic equation.

3*(x2−4*x−45)=0

3*(x−9)*(x+5)=0

x=9,x=−5

4. Find the second derivative to apply the Second Derivative Test for concavity.

d2(y)/(d(x)2)=6*x−12

5. Test the critical point x=9 in the second derivative.

d2(y)/(d(x)2)=6*(9)−12=42

Since 42>0 the graph is concave up, indicating a local minimum.

6. Test the critical point x=−5 in the second derivative.

d2(y)/(d(x)2)=6*(−5)−12=−42

Since −42<0 the graph is concave down, indicating a local maximum.

7. Calculate the y-coordinates by substituting the critical points back into the original function.

y(9)=(9)3−6*(9)2−135*(9)=729−486−1215=−972

y*(−5)=(−5)3−6*(−5)2−135*(−5)=−125−150+675=400

Final Answer

Local Maxima: *(−5,400), Local Minima: *(9,−972)

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