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Find the value of ‘a’ & ‘b’ if f(x) = ax2 + bx + sin(x) is continuous over and differentiable over (0, π) and satisfy the Rolle’s theorem at point c = π⁄4.

Find the value of ‘a’ & ‘b’ if f(x) = ax2 + bx + sin(x) is continuous over and differentiable over (0, π) and satisfy the Rolle’s theorem at point c = π⁄4. Correct Answer 0, π

Since function f(x) is continuous over and satisfy rolle’s theorem, ⇒ f(0) = f(π) = 0 ⇒ f(π) = a π2 + b π=0 ⇒ a π+b=0 ………………….(1) Since it satisfies rolle’s theorem at c = π⁄4 f’(c) = 2ac + b + Cos(c) = 0 ⇒ a(π⁄2) + b + 1⁄√2 = 0 ………………..(2) From eq(1) and eq(2) we get, ⇒ a = 0.45 ⇒ b = -1.414.

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