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# Up to concave down at x 0 f 0 0 and f x is positi

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Unformatted text preview: e must increasing be positi v e . We take the first derivative of f ’(x) which is f ”(x), the positiv a second derivative of f(x), and compute f ”(a). f ”(a ) > 0 implies a is minimum. a This is usually the easiest and most useful method. But there are exceptions. 117 1: Exercise 1: For the function f(x) = x 3 f ’(x) = 3x 2 . At x = 0 : f ’(0) = 0 implies zero is an extreme point. Is x = 0 a maximum point or minimum point ? Use all three methods to determine this. Exercise 2: For the following functions find the extreme points and determine if they are maximum or minimum. a) x + 2 b) x 2 c) -- x 2 d) x 3 e) -- x 3 f) x4 g) -- x 4 From the number of times a function intersects the x-axis (the number of real roots) can you estimate the number of extreme points ? Exer ercise Ex er cise 3: Divide 10 into two par ts such that the product is a maximum. Exer ercise Ex er cise 4: Divide 5 into two par ts such that the product is a maximum. Draw the graph and verify. Exer ercise Ex er cise 5: Divide 2 into two par ts such that the product is a maximum. Draw the graph and verify. 118 Absolute Maximum and Absolute Minimum It is possible to have more than one maximum and one minimum as in f(t) below. maximum minimum maxim minim f(t) 0 t0 t1 t2 t3 t4 t5 t6 t7 t8 t In this case we will find the overall or absolute maximum and the overall or absolute absolute minimum . By solving f(t) = 0 we find all the maximum and minimum maximum minimum points. Each maximum point is called a local maximum . And each minimum maximum maximum minimum point is called a local minimum . minimum absolute maximum = maximum (all local maximum ) maximum local maximum maximu absolute minimum = minimum (all local minimum ) minimum local minimum Exercise : Find the maximum and minimum points of f(t) = sin (+3ω0t) . a m p l i t u d e f(t) = sin (+3ω0t) T 119 At a maxim um minim um or inf lexion point, the tangent t o the curve is maximum minimum um, infle lexion tangent t ang PARALLEL to the x-axis and hence tan θ = 0 . C onversely, if the tang ent t o the tangent t ang cur ve at a point is...
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