關於Differentiation的簡單A.Maths.問題

Tieria

2009-08-23 4:12 am

For each of the following functions:
(a) determine the range of values of x that the function is
(i) increasing,
(ii) decreasing;
(b) find the maximum and minimum points.

1. y = x^2 – 2x – 3

dy / dx = 2x – 2

when dy / dx = 0,

2x – 2 = 0
x = 1

X

X < 1

X = 1

X > 1

Dy / dx

-

0

+

(a)(i) increasing: [1,∞]←後方括應為圓括

(ii) decreasing: (-∞, 1)←後圓括應為方括

(b) Minimum point: (1,- 4)

問題：在這條題目中，為何會有Minimum point: (1,- 4)？何以判斷？

回答 (1)

用戶9112

2009-08-23 5:58 am

✔ 最佳答案

Because when x changes from negative infinity up to 1, the curve is decreasing. This means the value of y decreases continuously when x changes from negative infinity gradually to x = 1. Then from x = 1 onwards to positive infinity the curve is increasing. Therefore the y value continously increases from the point x = 1 onwards. So x= 1 represents the minimum point.
Indeed the maximum and minimum points can be determined by equating dy/dx = 0. Then by evaluating the second derivative, we can conclude if the point is min or max. In this case, the second derivative = 2 which is positive. The point is then a min point.

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