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2 c Clearly by a and b above f is neither increasing nor decreasing in 0 2 Example 10 Find the intervals in which the function f given by f x x 4x 6 is a increasing b decreasing Solution We have or f x x 4x 6 f x 2x 4 156 Therefore f x 0 gives x 2 Now the point x 2 divides the real line into two disjoint intervals namely 2 and 2 Fig 6 3 In t 4 0 interval 2 f x 2x Therefore f is decreasing in this interval Also in the interval 2 f x 0 and so the function f is increasing in this interval Solution We have Example 11 Find the intervals in which the function fgiven by f x 4x 6x 72x 30 is a increasing b decreasing f x 4x 6x 72x 30 12x 12x 72 12 x x 6 12 x 3 x 2 Therefore f x 0 gives x 2 3 The points x 2 and x 3 ides the real line into three disjoint intervals namely 2 2 3 and 3 00 or 80 MATHEMATICS Interval 2 2 3 3 00 8 Rationalised 2023 24 Sign of f x 0 0 0 pun you Fig 6 4 3 In the intervals 2 and 3 f x is positive while in the interval 2 3 f x is negative Consequently the function is increasing in the intervals 2 and 3 0 while the function is decreasing in the interval 2 3 However fis neither increasing nor decreasing in R Example 12 Find intervals in which the function given by f x a increasing b decreasing Nature of function f fis increasing fis decreasing fis increasing o Sin 3x xe 0 1 is

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