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 negative second derivative indicates a local maximum and a positive second derivative indicates a local minimum.

Local Maximum at *x=0

Local Minima 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: *(0,18), Local Minima: *(√(,5),−32),(−√(,5),−32)

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