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Question

The function f:[0,3] [1,29], defined by f(x)=2x315x2+36x+1, is

A
one-one and onto.
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B
onto but not one-one.
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C
one-one but not onto.
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D
neither one-one nor onto.
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Solution

The correct option is B onto but not one-one.
f(x)=2x315x2+36x+1,f:[0,3][1,29]
1. Check for one-one mapping-
f(x) is a cubic polymonial. First derivative of f(x) gives f(x)=6x230x+36.
Solve f(x)=0 to get the critical points.
x25x+6=0
(x3)(x2)=0x{3,2}
f(x) is increasing for x<2 or x>3 (f(x)>0) and f(x) is decreasing for 2<x<3 (f(x)<0).
Hence, f(x) is not strictly increasing or strictly decreasing in the entire domain. So f(x) is not one-one.
2. Check for onto mapping-
f(x) has maximum at x=2
f(2)=29, which is maximum value of f.
( in the given domain [0,3], f is increasing in [0,2) and decreasing in (2,3].)
f(0)=1 and f(3)=28
Hence, range of f(x)=[1,29] which is equal to the co-domain. Hence, f(x) is onto.

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