Use the Second Derivative Test to Complete these problems. 1) Consider the funct

ID: 2857375 • Letter: U

Question

Use the Second Derivative Test to Complete these problems.

1) Consider the function f(x) =( (1/4)x^4)-((1/3)x^3)-(3x^2)+(3)

a. Find the x values of any points where the tangent line to f(x) is horizontal.

b. Use the Second Derivative Test to determine any relative extreme values.

c. Then complete the sentences below:

-At___x=____, the function f has a relative____ minimum______of value:

-At___x=____, the funnction f has a relative____ maximum______ of value:

-At___x=_____, the function f has a relative ____ minimum ______ of value:

given f(x) =( (1/4)x^4)-((1/3)x^3)-(3x^2)+(3)

for horizontal tangent line f '(x)=0, f '(x) changes its sign

f '(x) =( (1/4)4x3)-((1/3)3x2)-(3*2x)+0 =0

f '(x) =x3-x2-6x =0

x3-x2-6x=0

x(x2-x-6)=0

x(x-3)(x+2)=0

at x=-2,x=0,x=3 tangent line is horizontal

f '(x) =x3-x2-6x

differentiate

f ''(x) =3x2-2x-6

f "(-2)=12+4-6=10>0 ==> relative minimum

f "(0)=-6<0 ==> relative maximum

f "(3)=27-18-6=3>0 ==> relative minimum

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