Find the Local Maxima and Minima f(x)=5x^3-15x

Problem

ƒ(x)=5*x3−15*x

Solution

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

(d(5)*x3−15*x)/d(x)=15*x2−15

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

15*x2−15=0

3. Solve for x by factoring or using algebraic manipulation.

15*(x2−1)=0

x2=1

x=1,x=−1

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

(d(15)*x2−15)/d(x)=30*x

5. Test the critical points by substituting them into the second derivative.

ƒ(1)″=30*(1)=30

ƒ″*(−1)=30*(−1)=−30

6. Determine the nature of the points: since ƒ(1)″>0 there is a local minimum at x=1 since ƒ″*(−1)<0 there is a local maximum at x=−1

7. Calculate the y-coordinates by substituting the critical points back into the original function ƒ(x)

ƒ(1)=5*(1)3−15*(1)=−10

ƒ*(−1)=5*(−1)3−15*(−1)=10

Final Answer

Local Maxima: *(−1,10), Local Minima: *(1,−10)

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