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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)=−x2−3x+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)≥0⇒k≥0 h(8)>0⇒−64−24+k>0⇒k>88 h(9)<0⇒−81−27+k<0⇒k<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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