Q.39 The graph of the function
F(x) = x / (k1x2 + k2x + 1)
for 0 < x < β is
Correct Answer: (A)
Introduction
In this problem, we analyze the graph of the function
F(x) = xβk1x2 + k2x + 1
for positive values of x. By studying limits, critical points, and overall behavior,
we identify the correct graphical representation.
Step-by-Step Analysis
1. Behavior as x β 0+
When x approaches 0:
F(x) β xβ1 = x
Hence, the graph starts from the origin and increases.
2. Behavior as x β β
For large values of x:
F(x) β xβk1x2 = 1βk1x β 0
Thus, the function approaches zero asymptotically.
3. Maximum Point
Differentiating F(x):
F'(x) = (k1x2 + k2x + 1) β x(2k1x + k2)β (k1x2 + k2x + 1)2
Setting the numerator equal to zero:
βk1x2 + 1 = 0
x = 1ββk1
Hence, the function has a single maximum at this point.
Conclusion
The function increases from the origin, reaches a single maximum at
1ββk1,
and then decreases asymptotically to zero. Therefore,
Option (A) correctly represents the graph.
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