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Find the value of ‘a’ if f(x) = ax2+32x+4 is continuous over and differentiable over (-4, 0) and satisfy the Rolle’s theorem. Hence find the point in interval (-2,0) at which its slope of a tangent is zero

Find the value of ‘a’ if f(x) = ax2+32x+4 is continuous over and differentiable over (-4, 0) and satisfy the Rolle’s theorem. Hence find the point in interval (-2,0) at which its slope of a tangent is zero Correct Answer -4, 0

Since it satisfies Rolle’s Theorem, f’(c) = 0 = 2ac+32 ………………(1) and, f(0) = 4 hence by Rolle’s theorem and, f(-4) = 4 = 16a-128+4 (because f(0)=f(-4) condition of rolle’s theorem) ⇒ a = 8 from, eq.(1) ⇒ c = -2.

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