Let f x -sin x - ax - b - Then f x -0 hasA- only one real root which is positive if a-1- b1- b -0C- only one real root which is negative if a 0D- None of these-

The correct option is A only one real root which is positive if a gt; 1, b lt; 0 f'(x) = cosx + a, if a 1,then f(x) entirely increasing. So f(x) =0 ...

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Standard XII

Mathematics

Local Maxima

Let f x =sin ...

Question

Let f (x) = sinx + ax + b. Then f(x) = 0 has

only one real root which is positive if a > 1, b < 0

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only one real root which is negative if a > 1, b < 0

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only one real root which is negative if a < -1, b > 0

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None of these .

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Solution

The correct option is A only one real root which is positive if a > 1, b < 0
f'(x) = - cosx + a, if a $>$ 1,then f(x) entirely increasing. So f(x) =0 has only one real root, which is positive if f(0) $<$ 0 and negative if f(0) $>$ 0.

Similarly when a $<$ -1. Then f(x) entirely decreasing. So f(x) has only one real root which is negative if f(0) $<$ 0 and positive if f(0) $>$ 0

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