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數學：綜合解方程
Two real roots of the equation 17 - n² = eⁿ are denoted by α and β.
a) Show that the equation has exactly two real roots.
b) Find, by calculations, the pair of consecutive integers between which α+β lies.
1 個解答
- 知足常樂Lv 76 年前最愛解答
a)
It will be very clear by considering the graphs of
y = e^x and y = -x² + k by varying k.
Let f(x) = e^x and g(x) = 17 - x², both are continuous.
For x > 0, f(x) is increasing and g(x) is decreasing.
Also, g(0) = 17 > 1 = f(0).
There must be one root root of f(x) = g(x) for x > 0.
2015-06-13 22:16:54 補充：
f(x) > 0 for all x (then all x < 0)
g(x) < 0 for all x < -√17
Again with g(0) = 17 > 1 = f(0) and their continuity and monotonicity.
There must be exactly one root of f(x) = g(x) for x < 0.
Combining, there are exactly two roots of f(x) = g(x).
2015-06-22 00:49:09 補充：
By calculations, the two roots are -4.12114 and 2.41363.
The sum is -4.12114 + 2.41363 = -1.70751.
Thus, the pair of consecutive integers required are -1 and -2.
2015-06-24 16:50:19 補充：
a)
It will be very clear by considering the graphs of
y = e^x and y = -x² + k by varying k.
Let f(x) = e^x and g(x) = 17 - x², both are continuous.
For x > 0, f(x) is increasing and g(x) is decreasing.
Also, g(0) = 17 > 1 = f(0).
There must be one root root of f(x) = g(x) for x > 0.
f(x) > 0 for all x (then all x < 0)
g(x) < 0 for all x < -√17
Again with g(0) = 17 > 1 = f(0) and their continuity and monotonicity.
There must be exactly one root of f(x) = g(x) for x < 0.
Combining, there are exactly two roots of f(x) = g(x).
b)
By calculations, the two roots are -4.12114 and 2.41363.
The sum is -4.12114 + 2.41363 = -1.70751.
Thus, the pair of consecutive integers required are -1 and -2.
