ANANT SIR OR ANY EXPERT SIR PLEASE solve the question If the function integral(

FORESTS

solve the question If the function integral(f(t)dt from,0 to x) approaches to5 as |x| approaches to 1 then the value of 2x+ integral(f(t)dt from 0 to x)=a has at least two roots of opposite sign in (-1,1)is

Srujana

$\int_{0}^{x} f(t).dt \rightarrow 5$ as $|x|$ $\rightarrow 1$

$\int_{0}^{-1} f(t).dt \rightarrow 5$

$\int_{0}^{1} f(t).dt \rightarrow 5$

$g(x)=2x+{\int_{0}^{x} f(t).dt} -a$

If $g(x)$ has atleast two roots of opposite signs in $(-1,1)$

$g(-1).g(1)>0$ $\Rightarrow (a-3)(a-7)>0$ $\Rightarrow a \in(\infty,3) \cup (7,\infty)$

$g \left(0\right).g(-1)<0$ $\Rightarrow (a)(a-3)<0$ $\Rightarrow a \in(0,3)$

$g \left(0\right).g(1)<0$ $\Rightarrow (a)(a-7)<0$ $\Rightarrow a \in(0,7)$

$(0,3) \cap (0,7) \cap [(\infty,3) \cup (7,\infty)]$

$\Rightarrow a\in (0,3)$

FORESTS

thankyou very much

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