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Application of Integration question

2024 · Shift 2 · Q30
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  5. /2024 · Shift 2 · Q30

Application of Integration question

2024 · Shift 2 · Q30

JEE AdvancedMathematicsApplication of IntegrationNumerical+4 / −1
Let the function f:[1,∞)→Rf:[1, \infty) \rightarrow \mathbb{R}f:[1,∞)→R be defined by f(t)={(−1)n+1⋅2if t=2n−1,n∈N(2n+1−t)2f(2n−1)+(t−(2n−1))2f(2n+1)if 2n−1<t<2n+1,n∈Nf(t)=\left\{\begin{array}{ll} (-1)^{n+1} \cdot 2 & \text{if } t=2n-1, n \in \mathbb{N} \\ \dfrac{(2n+1-t)}{2} f(2n-1)+\dfrac{(t-(2n-1))}{2} f(2n+1) & \text{if } 2n-1 \lt t \lt 2n+1, n \in \mathbb{N} \end{array}\right.f(t)={(−1)n+1⋅22(2n+1−t)​f(2n−1)+2(t−(2n−1))​f(2n+1)​if t=2n−1,n∈Nif 2n−1<t<2n+1,n∈N​ Define g(x)=∫1xf(t) dtg(x)=\int_1^x f(t) \, dtg(x)=∫1x​f(t)dt, x∈(1,∞)x \in (1, \infty)x∈(1,∞). Let α\alphaα denote the number of solutions of the equation g(x)=0g(x)=0g(x)=0 in the interval (1,8](1,8](1,8] and β=lim⁡x→1+g(x)x−1\beta=\lim\limits_{x \rightarrow 1^+} \dfrac{g(x)}{x-1}β=x→1+lim​x−1g(x)​. Then the value of α+β\alpha+\betaα+β is equal to ‾\underline{\hspace{2cm}}​.
Numerical answer
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