Find the Local Maxima and Minima f(x)=x^4-18x^2+1

Problem

ƒ(x)=x4−18*x2+1

Solution

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

d(ƒ(x))/d(x)=4*x3−36*x

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

4*x3−36*x=0

3. Factor the expression to solve for the critical values of x

4*x*(x2−9)=0

4*x*(x−3)*(x+3)=0

x=0,x=3,x=−3

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

d2(ƒ(x))/(d(x)2)=12*x2−36

5. Evaluate the second derivative at each critical point to determine if it is a maximum or minimum.

ƒ(0)″=12*(0)2−36=−36

ƒ(3)″=12*(3)2−36=72

ƒ″*(−3)=12*(−3)2−36=72

6. Interpret the results based on the sign of the second derivative. Since ƒ(0)″<0 there is a local maximum at x=0 Since ƒ(3)″>0 and ƒ″*(−3)>0 there are local minima at x=3 and x=−3

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

ƒ(0)=(0)4−18*(0)2+1=1

ƒ(3)=(3)4−18*(3)2+1=81−162+1=−80

ƒ*(−3)=(−3)4−18*(−3)2+1=81−162+1=−80

Final Answer

Local Maxima: *(0,1), Local Minima: *(3,−80),(−3,−80)

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