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Question

If f(x)=x2+14x+9x2+2x+3, g(x)=x24x+9 and h(x)=3x24x+p,

then which of the following option(s) is/are correct?

A
Number of integers in the range of f(x) is 8
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B
Maximum value of g(f(x)) is 54
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C
Number of integral values of p for which h(f(x))>0 and h(g(x))<0 is 28
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D
Number of integral values of p for which h(f(x))>0 and h(g(x))<0 is 30
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Solution

The correct options are
B Maximum value of g(f(x)) is 54

D Number of integral values of p for which h(f(x))>0 and h(g(x))<0 is 30
y=x2+14x+9x2+2x+3
x2(y1)+(2y14)x+3y9=0
(2y14)24(y1)(3y9)0
y[5,4]

g(x)=x24x+9=(x2)2+5
Maximum value of g(f(x)) will occur at f(x)=5 and g(5)=54

Now, h(5)>0
3(5)24(5)+p>0p>55

Also,h(4)>03(4)24(4)+p>0p>64
and h(5)<03(5)24(5)+p<0p<95

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