Consider the following function. Consider the following function. f(x) x2 16 (a)

Question

Consider the following function.

Consider the following function. f(x) x2 16 (a) Find the critical numbers of f. Enter your answers as a comma-separated list.) X- 0,4 (b) Find the open intervals on which the function is increasing or decreasing. (Select all that apply.) Increasing: (-00, -4) (-4, 0) (0, 4) (4, oo) none of these Decreasing: (-00, -4) (-4, 0) (0, 4) OO none of these (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) (x, y) 00 relative maximum (x, y) ne relative minimum

Explanation / Answer

f(x)=x2/(x2-16)

doamin =(-,-4)U(-4,4)U(4,)

f '(x)=[2x(x2-16) -x2(2x-0)]/(x2-16)2

f '(x)=[2x3-32x -2x3]/(x2-16)2

f '(x)=-32x/(x2-16)2

for critical points f '(x)=0

-32x/(x2-16)2=0

x=0

increasing when f '(x)>0

-32x/(x2-16)2>0

x<0

x=(-,-4)U(-4,0)

decreasing when f '(x)<0

-32x/(x2-16)2<0

x>0

x=(0,4)U(4,)

c) relative maximum at x=0

f(0)=0

(x,y)=(0,0)

relative minimum DNE

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