JEE Advance - Mathematics (1997 - No. 21)
If $${{dg} \over {dx}} > 0$$ for all $$x$$, prove that $$\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} } $$
increases as $$(b-a)$$ increases.
increases as $$(b-a)$$ increases.
The question is about showing the relationship between the change of the difference of two variables (b-a), where a+b is constant, to the increase of the value of two integral functions with g(x) as their integrands.
The condition $$\frac{dg}{dx} > 0$$ implies that g(x) is an increasing function.
Because g(x) is an increasing function and a+b is constant, increasing b-a is equivalent to increasing b and decreasing a.
Increasing b will make the integral with the range (0,b) smaller.
The overall change depends on the properties of g(x) and the values of a and b; it can decrease, stay the same or increase.
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