Find the Local Maxima and Minima G(x)=-2x^4+20x^2-18

Problem

G(x)=−2*x4+20*x2−18

Solution

1. Find the first derivative of the function to identify critical points.

d(G(x))/d(x)=−8*x3+40*x

2. Set the derivative to zero and solve for x to find the critical values.

−8*x3+40*x=0

−8*x*(x2−5)=0

x=0,x=√(,5),x=−√(,5)

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

d2(G(x))/(d(x)2)=−24*x2+40

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

G(0)″=−24*(0)2+40=40

G(√(,5))″=−24*(5)+40=−80

G″*(−√(,5))=−24*(5)+40=−80

5. Interpret the results where a positive second derivative indicates a local minimum and a negative second derivative indicates a local maximum.

Local Minimum at *x=0

Local Maxima at *x=√(,5),x=−√(,5)

6. Calculate the y-coordinates by substituting the critical values back into the original function G(x)

G(0)=−2*(0)4+20*(0)2−18=−18

G(√(,5))=−2*(25)+20*(5)−18=32

G*(−√(,5))=−2*(25)+20*(5)−18=32

Final Answer

Local Maxima: *(±√(,5),32), Local Minimum: *(0,−18)

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