lecture_4 - Lecture 4 Optimisation Slope of a curve and...

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Lecture 4 – Optimisation Slope of a curve and turning points - at a turning point, the slope of a curve is always zero For y = f(x), to find a turning point Step 1: Find dy/dx for a given curve y =f(x) Step 2: Solve the equation dy/dx = 0 There can be local or global maxima, and local and global minima ECON Lecture 4 1
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ECON Lecture 4 2 Determining maximum and minimum turning points Note - when y is increasing, y’ is positive - when y’ is increasing y’’ is positive - when y is decreasing y’ is negative - when y’ is decreasing y’’ is negative From this it follows that: - when y is increasing dy/dx > 0 - when dy/dx is increasing , then d 2 y/dx 2 > 0 - when dy/dx is decreasing dy/dx < 0 - when dy/dx is decreasing, d 2 y/dx 2 < 0
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ECON Lecture 4 3 Test for a maximum At the maximum x 0 , the slope changes from positive to negative - the slope must be zero at the maximum Method 1: Evaluate the slope dy/dx at values immediately before and after - if the slope changes from positive (before) to negative (after) the point is a maximum Method 2:
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