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Question

# Let f:R→R be defined by f(x)=x2−3x−6x2+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)=x2−3x−6x2+2x+4 ⇒f′(x)=(x2+2x+4)(2x−3)−(x2−3x−6)(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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