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DOON INTRNATIONAL SCHOOL YEAR 2020 - 2021

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I,  Anmol  Ahluwalia ,  student of class XII  C  would like to express my gratitude towards my teacher Mr. Parag Sharma and my friends  for helping me in preparing this presentation, my Mathematics  AIL  activity.

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For a given function, y = f(x), if the value of y is increasing on increasing the value of x, then the function is known as an increasing function and if the value of y is decreasing on increasing the value of x, then the function is known as a decreasing function.

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Let y = f(x) be a differentiable function on an interval (a, b). If for any two points x1, x2 ∈ (a, b) such that x1 < x2 there holds the inequality then f(x1) ≤ f(x2), the function is called increasing (or non-decreasing) in this interval.

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Let y = f(x) be a differentiable function on an interval (a, b). If for any two points x1, x2 ∈ (a, b) such that x1 < x2 there holds the inequality then f(x1) ≥ f(x2), the function is called decreasing(or non-increasing) in this interval.

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Let f(x)=2x  3  −9x  2  +12x+15 f ′ (x)=6x  2  −18x+12 =6(x  2  −3x+2) =6(x−1)(x−2) f ′ (x)=0 gives us 6(x−1)(x−2)=0 ⇒x=1,2 The points x= 1,2 divide the real line into three intervals (−∞,1),(1,2),(2,∞)

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1. In the interval(−∞,1),  f ′ (x)>0 ∴f(x) is increasing in(−∞,1) 2. In the interval (1,2),  f ′ (x)<0 ∴f(x)is decreasing in(1,2) 3. In the interval (2,∞),  f ′ (x)>0 ∴f(x)is increasing in(2,∞).

MATHEMATICS AIL PROJECT