Find the Local Maxima and Minima f(x)=x^4-4x^3+10

Problem

ƒ(x)=x4−4*x3+10

Solution

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

d(ƒ(x))/d(x)=4*x3−12*x2

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

4*x3−12*x2=0

3. Factor the expression to solve for the variable x

4*x2*(x−3)=0

4. Identify the critical values by solving the factored equation.

x=0

x=3

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

d2(ƒ(x))/(d(x)2)=12*x2−24*x

6. Test the critical point x=3 by substituting it into the second derivative.

ƒ(3)″=12*(3)2−24*(3)

ƒ(3)″=108−72=36

Since ƒ(3)″>0 the function is concave up, indicating a local minimum at x=3

7. Test the critical point x=0 by substituting it into the second derivative.

ƒ(0)″=12*(0)2−24*(0)=0

Since the Second Derivative Test is inconclusive, use the First Derivative Test. For x<0 ƒ(x)′<0 For 0<x<3 ƒ(x)′<0 Since the sign does not change, x=0 is an inflection point, not a local extremum.

8. Calculate the y-coordinate for the local minimum by substituting x=3 into the original function.

ƒ(3)=(3)4−4*(3)3+10

ƒ(3)=81−108+10=−17

Final Answer

Local Minimum: *(3,−17), Local Maxima: None

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