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  • Given <img alt="\mathrm{\left|\int_a^b f(x) d x\right|=\int_a^b|f(x)| d x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cleft%7C%5Cint_a%5Eb%20f%28x%29%20d%20x%5Cright%7C%3D%

Given \mathrm{\left|\int_a^b f(x) d x\right|=\int_a^b|f(x)| d x} and \mathrm{f^{\prime}(x) \neq 0} at any \mathrm{x \in(a, b)}. If \mathrm{g(x)=\int_0^x f(t) d t, f(x)} being continuous and differentiable in \mathrm{(a, b)}; then \mathrm{\int_a^b f(x) g(x) d x=0} implies

Option: 1

\mathrm{g(x)=0 \, \, has \, \, atmost\, \, one \, \, root \, \, in (a, b) root \, \, in (a, b)}


Option: 2

\mathrm{g(x)=0 \text { has atleast one }}


Option: 3

\mathrm{g(x)=0 \text { has exactly one root in }(a, b)}


Option: 4

\mathrm{g(x)=0 \quad \forall x \in(a, b)}


Answers (1)

best_answer

\mathrm{g^{\prime}(x)=f(x)} which is either positive or negative in (a, b)

\mathrm{\Rightarrow g(x)} is either increasing or decreasing in (a, b)

\mathrm{\therefore \int_a^b f(x) g(x) d x=0 \Rightarrow g(x)} must go from positive to negative or from negative to positive.

\mathrm{\Rightarrow g(x)=0 \text { has exactly one root in }(a, b) \text {. }}

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Rishi

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