Passage: Consider the curve C_1 given by y = e^ -x for x [0, 10 ], and the curve C_2 given by y = e^ -x ( x + x) for x [0, 10 ]. Let n be the total number of points of intersection of the curves C_1 and C_2. Suppose that _1, _2, , _n [0, 10 ] are the x-coordinates of the points of intersection of the curves C_1 and C_2 such that _1 Question: The value of n is ___________.
A. A
Equating the equations of the curves C_1 and C_2: e^ -x = e^ -x ( x + x) Since e^ -x 0 for all real x, dividing both sides by e^ -x gives: x + x = 1 Multiplying both sides by 1 2 : 1 2 x + 1 2 x = 1 2 (x + 4 ) = 1 2 The general solution is given by: x + 4 = 2m + 4 or x + 4 = 2m + 3 4 for integer m This simplifies to: x = 2m or x = 2m + 2 Given the interval x [0, 10 ], the valid values for x are: For x = 2m : x = 0, 2 , 4 , 6 , 8 , 10 (6 solutions) For x = 2m + 2 : x = 2 , 5 2 , 9 2 , 13 2 , 17 2 (5 solutions) The total number of points of intersection is n = 6 + 5 = 11. Answer: 11
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