Maxima and minima - Maxima and minima Increasing and...

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Maxima and minima
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Increasing and Decreasing Functions An increasing function is one whose gradient is always greater than or equal to zero. for all values of x A decreasing function has a gradient that is always negative or zero. for all values of x 0 dx dy 0 dx dy
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Increasing and Decreasing Functions A STRICTLY increasing function is one whose gradient is always greater than or equal to zero. for all values of x A STRICTLY decreasing function has a gradient that is always negative or zero. for all values of x 0 dx dy 0 dx dy
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e.g.1 Show that is an increasing function Solutio n: a positive number ( 3 ) a perfect square (which is positive or zero for all values of x ) , and for all values of x is the sum of a positive number ( 4 ) so, is a strictly increasing function 4 3 2 x dx dy x x y 4 3 x x y 4 3 0 dx dy dx dy x x y 4 3
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Solutio n: e.g.2 Show that is an increasing function. To show that is never negative is an increasing function. for all values of x Since a square is always greater than or equal to zero, 9 6 2 x x dx dy x x x y 9 3 2 3 3 1 x x x y 9 3 2 3 3 1 9 6 2 x x 2 2 ) 3 ( 9 6 x x x x x x y 9 3 2 3 3 1 0 dx dy
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The graphs of the increasing functions and are an d x x y 4 3 x x x y 9 3 2 3 3 1 x x y 4 3 x x x y 9 3 2 3 3 1
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Exercise s 1. Show that is a decreasing function and sketch its graph. 2. Show that is an increasing function and sketch its graph 3 x y
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1. Show that is a decreasing function and sketch its graph. Solution s Solution: . This is the product of a square which is always and a negative number, so for all x . Hence is a decreasing function. 3 x y 2 3 x dx dy 0 dx dy 0 3 x y 3 x y
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Solution s 2. Show that is an increasing function and sketch its graph , and therefore thus the function is increasing
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Stationary Points Maxima and minima
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The stationary points of a curve are the points where the gradient is zero A local maximum A local minimum x x The word “local” is usually left out and the points are just called maximum and minimum points e.g. x x x y 9 3 2 3 0 dx dy
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e.g.1 Find the coordinates of the stationary points on the curve Solutio n: o r The stationary points are (3, -27) and ( -1, 5) Tip: Watch out for common factors when finding stationary points.
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