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Question

Let f, g and h be three functions defined as follows :
f(x)=324+x2+x4, g(x)=9+x2 and h(x)=x23x+k.
Which of the following statement(s) is(are) CORRECT?

A
Number of integers in the range of f(x) is 8.
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B
Number of integral values of k for which h(f(x))>0 and h(g(x))<0 xR is 20.
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C
Number of integral values of k for which h(f(x))>0 and h(g(x))<0 xR is 19.
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D
Maximum value of g(f(x)) is 73.
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Solution

The correct option is D Maximum value of g(f(x)) is 73.
f(x)=324+x2+x4
Clearly, domain of f is R.
Since x4+x2+4[4,),
Range of f is (0,8]

h(f(x))>0 and h(g(x))<0
Since f(x)(0,8] and g(x)[9,),
h(0)0k0
h(8)>06424+k>0k>88
h(9)<08127+k<0k<108
Hence, k{89,90,,106,107}
Number of integral values of k is 19.

Since f(x)(0,8],
Maximum value of g(f(x)) is g(8)=64+9=73

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