Question

If ƒ is a continuous function such that $\int_{0}^{x}{ƒ(t)}$dt® as |x| ®, then for all k Ī R, equation k²x² + $\int_{0}^{x}{ƒ(t)}$dt – a = 0 (a > 0) has–

• A. All roots in (–, 0)
• B. All roots in (0, )
• C. Odd number of roots in (–, 0) and odd number of roots in (0, )
• D. None of these

Answer 1

Answer: (C)

Odd number of roots in (–, 0) and odd number of roots in (0, )

Answer 2

Let g(x) = k²x² +$\int_{0}^{x}{f(t)dt - a}$ … (1)

Since, a > 0 and $\int_{0}^{x}{f(t)dt \rightarrow \infty}$ as x ® ±  (given)

Ž g(0) = 0 + $\int_{0}^{0}{f(t)dt - a}$= –a < 0

and g() =  +  – a =  > 0

also, g(–) =  +  – a =  > 0

Ž g(x) is continuous in (–, )

Hence, g(x) = 0 has odd number of roots in (–, 0) and odd number of roots in (0, )