alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Polynomials of degree 012 and 3 are sufficient to

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Unformatted text preview: : f (a -- δ x) < f(a) and f (a+ δ x) < f(a) a a a a implying that a is a maximum point or f (a -- δx) > f(a) and f (a+δx) > f(a) a a a a implying that a is a minimum point. These calculations may be tedious. If we know the graph of the function then we can determine whether a is minimum. maximum or minimum This Geometric m ethod may not be practical. So let us try to find an Analytical method. Analytical Anal We shall use the results : if f(x) is increasing then f ’(x) is positive positive. if f ’(x) is positive then f(x) is increasing increasing. if f(x) is decreasing then f ’(x) is negative negative. if f ’(x) is negative then f(x) is decreasing . decreasing. 116 Method 2 : At the maximum point a : f(x) is increasing before and after. decr easing after. Correspondingly, f ’(x) is positi v e befor e and ne g a ti v e after. positiv befor ore neg tiv after. We need to check around a if : f ’(a -- δx) > 0 and f ’(a + δ x) < 0. a a At the minimum point a : f(x) is decreasing before and after. incr easing after. Correspondingly, f ’(x) is ne g a ti v e befor e and positi v e after. neg tiv befor ore positiv after. We need to check around a if : f ’(a -- δx) < 0 and f ’(a + δ x) > 0. a a This again may be tedious. maximum positiv befor ore Method 3: We know that at the maxim um point a : f ’(x) is positi v e befor e and ne g a ti v e after. neg tiv after. Going from positive to negative means f ’(x) must be decreasing. We also know that if a function is decr easing its first derivative decreasing must be ne g a ti v e . We take the first derivative of f ’(x) which is f ”(x), neg tiv the second derivative of f(x), and compute f ”(a). a f ”(a) < 0 implies a is maximum. a Similarly, we know that at the minimum point a : f ’(x) is negative befor ore positiv after. bef or e and positi v e after. Going from negative to positive means f ’(x) must be increasing. We also know that if a function is incr easing its first derivativ...
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