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數學：綜合解方程
A real root of the equation 17 - x² = 6 ln x is denoted by α.
a) Show that the equation has exactly one real root.
b) Find, by calculations, the pair of consecutive integers between which α lies.
2 個解答
- 知足常樂Lv 76 年前最愛解答
a)
Let f(x) = x² + 6 ln x - 17 for x > 0
The equation is f(x) = 0.
Consider f'(x) = 2x + 6/x > 0 for x > 0
f(x) is strictly increasing and continuous.
f(x) reaches the level of 0 exactly once.
Thus, there is exactly one real root for f(x) = 0.
2015-06-13 21:27:38 補充：
b)
f(1) = -16
f(2) = -8.841116917
f(3) = -1.408326268
f(4) = 7.317766167
3 < α < 4.
2015-06-24 16:48:13 補充：
a)
Let f(x) = x² + 6 ln x - 17 for x > 0
The equation is f(x) = 0.
Consider f'(x) = 2x + 6/x > 0 for x > 0
f(x) is strictly increasing and continuous.
f(x) reaches the level of 0 exactly once.
Thus, there is exactly one real root for f(x) = 0.
b)
f(1) = -16
f(2) = -8.841116917
f(3) = -1.408326268
f(4) = 7.317766167
3 < α < 4.
