Let h(x)=f(x)−a(f(x))2+a(f(x))3 for every real number x. h(x) increases as f(x) increases for all real values of x if
A
a∈(0,4)
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B
a∈(−2,2)
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C
a∈[3,∞)
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D
None of these
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Solution
The correct option is D None of these h(x)=f(x)−a(f(x))2+a(f(x))3 or h′(x)=f′(x)−2af(x)f′(x)+3a(f(x))2f′(x) =f′(x)[3a(f(x))2−2af(x)+1] Now, h(x) increases as f(x) increases when, 3a(f(x))2−2af(x)+1>0∀x∈R 3a>0 and 4a2−12a≤0 a>0 and a∈[0,3] So a∈[0,3]