Consider the following function. f(x)=frac{x^{2}}{x^{2}-81} a) To find the critucal numbers of f. b) To find the open interval on which function is increasing or decreasing. c) To identify the relative extremum.

waigaK 2020-10-18 Answered
Consider the following function. f(x)=x2x281 a) To find the critucal numbers of f. b) To find the open interval on which function is increasing or decreasing. c) To identify the relative extremum.
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SoosteethicU
Answered 2020-10-19 Author has 102 answers

a) Let us find first derivative of the function f. f(x)=(x281)2xx2(2x)(x281)2=2x3162x2x3(x281)2=162x(x281)2 We know that critical numbers are those number where the derivative vanishes or does not exist. f(x)=0x=0 Clearly f(x) does not exist when x281=0x=±9 Thus the critical numbers are - x=0,9,9 b) For increasing, f(x)>0
162x(x281)2>0
162x>0
x<0
x(,0) b) For decreasing f(x)<0
162x(x281)2<0162x<0
x>0
x(0,) Thus, increasing on (,0) decreasing on (0,) c) By first derivative test, f(x)<0 on (0,)
andf(x)>0 on (,0)
x=0 gives relative maximum. y=0 and relative minimum does not exist. Therefore, relative maximum (x,y)=(0,0) relative minimum (x,y)=DNE

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