# HW4 - Section 4.1 13 f x = 3 The derivative is positive for...

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Unformatted text preview: Section 4.1 13. f ( x ) = 3 . The derivative is positive for all x so that the function is increasing on (-∞ , + ∞ ) and the function is never decreasing. 15. f ( x ) = 2 x- 3 = 0-→ x = 3 / 2 is the only critical number at which f ( x ) = 0. There are no critical numbers at which f ( x ) is undefined. Evaluating f ( x ) at test points on either side of the critical number we find f (0) =- 3 < 0 and f (2) = 1 > . Thus the function is decreasing on (-∞ , 3 / 2) and increasing on (3 / 2 , + ∞ ) . 17. g ( x ) = 1- 3 x 2 = 0-→ x = ± q 1 / 3 = ± ( √ 3 / 3) are the two critical numbers at which g ( x ) = 0. There are no critical numbers at which g ( x ) is undefined. Evaluating g ( x ) at test points on either side of each critical number we find g (- 1) =- 2 < , g (0) = 1 > , and g (1) =- 2 < . Thus the function is decreasing on (-∞ ,- q 1 / 3) ∪ ( q 1 / 3 , + ∞ ) and increasing on (- q 1 / 3 , q 1 / 3) . 19. g ( x ) = 3 x 2 + 6 x = 3 x ( x + 2) = 0-→ x =- 2 , 0 are the two critical numbers at which g ( x ) = 0. There are no critical numbers at which g ( x ) is undefined. Evaluating g ( x ) at test points on either side of each critical number we find g (- 3) = 9 > , g (- 1) =- 3 < , and g (1) = 9 > . Thus the function is decreasing on (- 2 , 0) and increasing on (-∞ ,- 2) ∪ (0 , + ∞ ) . 21. f ( x ) = x 2- 6 x + 9 = ( x- 3) 2 = 0-→ x = 3 is the only critical number at which f ( x ) = 0. There are no critical numbers at which f ( x ) is undefined. Evaluating f ( x ) at test points on either side of the critical number we find f (2) = 1 > 0 and f (4) = 1 > . Thus the function is increasing on (-∞ , 3) ∪ (3 , + ∞ ) and the function is never decreasing. 23. h ( x ) = 4 x 3- 12 x 2 = 4 x 2 ( x- 3) = 0-→ x = 0 , 3 are the two critical numbers at which h ( x ) = 0. There are no critical numbers at which h ( x ) is undefined. Evaluating h ( x ) at test points on either side of each critical number we find h (- 1) =- 16 < , h (1) =- 8 < , and h (4) = 64 > . Thus the function is decreasing on (-∞ , 0) ∪ (0 , 3) and increasing on (3 , + ∞ ) . 25. f ( x ) =- 1 ( x- 2) 2 . The function has no critical numbers at which the derivative is equal to zero, however, there is a critical number at x = 2 at which f ( x ) is undefined. Evaluating f ( x ) at test points on either side of the critical number we find f (1) =- 1 < 0 and f (3) =- 1 < . Thus the function is decreasing on (-∞ , 2) ∪ (2 , + ∞ ) and the function is never increasing. 27. h ( t ) =- 1 ( t- 1) 2 . The function has no critical numbers at which the derivative is equal to zero, however, there is a critical number at t = 1 at which h ( t ) is undefined. Evaluating h ( t ) at test points on either side of the critical number we find h (0) =- 1 < 0 and h (2) =- 1 < . Thus the function is decreasing on (-∞ , 1) ∪ (1 , + ∞ ) and the function is never increasing....
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HW4 - Section 4.1 13 f x = 3 The derivative is positive for...

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