121616949-math.112 - Figure 5.4 The sine and cosine...

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98 Chapter 5 Curve Sketching positive just to the right. If the derivative exists near a but does not change from positive to negative or negative to positive, that is, it is postive on both sides or negative on both sides, then there is neither a maximum nor minimum when x = a . See the first graph in figure 5.1 and the graph in figure 5.2 for examples. EXAMPLE 5.4 Find all local maximum and minimum points for f ( x ) = sin x + cos x using the first derivative test. The derivative is f ± ( x ) = cos x - sin x and from example 5.3 the critical values we need to consider are π/ 4 and 5 π/ 4. The graphs of sin x and cos x are shown in figure 5.4 . Just to the left of π/ 4 the cosine is larger than the sine, so f ± ( x ) is positive; just to the right the cosine is smaller than the sine, so f ± ( x ) is negative. This means there is a local maximum at π/ 4. Just to the left of 5 π/ 4 the cosine is smaller than the sine, and to the right the cosine is larger than the sine. This means that the derivative f ± ( x ) is negative to the left and positive to the right, so f has a local minimum at 5 π/ 4. π 4 5 π 4 . . . .. . . . . . ..............................................................................................
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Unformatted text preview: . .. . . . . . . . . . . . . .. . . .............................................................................................. . . . .. . . . . . ............................................... .. . . . . . . . . . . . . . . .. . . .............................................................................................. . . . .. . . . . . . . . . . . . . .. .............................................. Figure 5.4 The sine and cosine. Exercises In 1–13, find all critical points and identify them as local maximum points, local minimum points, or neither. 1. y = x 2-x ⇒ 2. y = 2 + 3 x-x 3 ⇒ 3. y = x 3-9 x 2 + 24 x ⇒ 4. y = x 4-2 x 2 + 3 ⇒ 5. y = 3 x 4-4 x 3 ⇒ 6. y = ( x 2-1) /x ⇒ 7. y = 3 x 2-(1 /x 2 ) ⇒ 8. y = cos(2 x )-x ⇒ 9. f ( x ) = (5-x ) / ( x + 2) ⇒ 10. f ( x ) = | x 2-121 | ⇒ 11. f ( x ) = x 3 / ( x + 1) ⇒ 12. f ( x ) = x 2 sin(1 /x ) x ± = 0 x = 0 13. f ( x ) = sin 2 x ⇒ 14. Find the maxima and minima of f ( x ) = sec x . ⇒...
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