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Question

Let f:RR be defined by f(x)=x23x6x2+2x+4. Then which of the following statements is(are) TRUE?

A
f is decreasing in the interval (2,1)
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B
f is increasing in the interval (1,2)
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C
f is onto
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D
Range of f is [32,2]
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Solution

The correct option is B f is increasing in the interval (1,2)
f(x)=x23x6x2+2x+4
f(x)=(x2+2x+4)(2x3)(x23x6)(2x+2)(x2+2x+4)2
f(x)=5x(x+4)(x2+2x+4)2
Sign scheme of f(x):
Hence, f is decreasing in the interval (2,1)

Hence, f is increasing in the interval (1,2)

f(x) has local maximum at x=4 and minimum at x=0
f(4)=116, f(0)=32 and
limx±f(x)=1
Hence, range of f(x) is [32,116]

Hence, f is not onto
As we can see in the figure:


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