Find the Local Maxima and Minima f(x)=3x^4-6x^2+7

Problem

ƒ(x)=3*x4−6*x2+7

Solution

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

d(ƒ(x))/d(x)=12*x3−12*x

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

12*x3−12*x=0

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

12*x*(x2−1)=0

12*x*(x−1)*(x+1)=0

x=0,x=1,x=−1

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

d2(ƒ(x))/(d(x)2)=36*x2−12

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

ƒ(0)″=36*(0)2−12=−12

ƒ(1)″=36*(1)2−12=24

ƒ″*(−1)=36*(−1)2−12=24

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(1)″>0 there is a local minimum at x=1
   Since ƒ″*(−1)>0 there is a local minimum at x=−1

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

ƒ(0)=3*(0)4−6*(0)2+7=7

ƒ(1)=3*(1)4−6*(1)2+7=4

ƒ*(−1)=3*(−1)4−6*(−1)2+7=4

Final Answer

Local Maxima: *(0,7), Local Minima: *(1,4),(−1,4)

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