Consider the following function. f(x)=x2x2−81 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.

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SoosteethicU · Accepted Answer

a) Let us find first derivative of the function f. f′(x)=(x2−81)2x−x2(2x)(x2−81)2=2x3−162x−2x3(x2−81)2=−162x(x2−81)2 We know that critical numbers are those number where the derivative vanishes or does not exist. f′(x)=0⇒x=0 Clearly f′(x) does not exist when x2−81=0⇒x=±9 Thus the critical numbers are - x=0,−9,9 b) For increasing, f′(x)&amp;gt;0⇒−162x(x2−81)2&amp;gt;0⇒−162x&amp;gt;0⇒x&amp;lt;0∴x∈(−∞,0) b) For decreasing f′(x)&amp;lt;0⇒−162x(x2−81)2&amp;lt;0⇒−162x&amp;lt;0⇒x&amp;gt;0∴x∈(0,∞) Thus, increasing on (−∞,0) decreasing on (0,∞) c) By first derivative test, ∴f′(x)&amp;lt;0 on (0,∞)andf′(x)&amp;gt;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

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.

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