State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x^3 – x^2 – 6x + 2 in [0, 3]

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State Rolle&#8217;s theorem and verify the Rolle&#8217;s theorem for f(x) = x^3 &#8211; x^2 &#8211; 6x + 2 in [0, 3]

Suresh Chand · Accepted Answer

Definition: Let f be a function that satisfies the following three hypothesis: f is continuous on the close interval [a, b] f is differentiable on the open interval (a, b) f(a) = f(b) Then there is a number c in (a, b) such that f'(c) = 0   Problem Part: Solution: Given f(x) = x3 – x2 – 6x + 2 on [0, 3] since, (lim_{x to 0}f(x) = 2) and (lim_{x to 3^{-}}f(x) = 2) Thus, f(x) is continuous at end points and all interior points of [0, 3] f(x) is continuous on the close interval [0, 3] f'(x) = 3x2 - 2x - 6 exists in (0, 3). So, f(x) is differentiable in (0, 3). f(0) = f(3) = 2 Thus, all three conditions are holds on f(x). Hence, there exists  point (c in (0, 3)) such that f'(c) = 0 or, 3c2 - 2c - 6 = 0 (or, c = frac{1 pm sqrt{19}}{3}) (or, c = frac{1 + sqrt{19}}{3} in (0, 3)) So, Rolle's theorem is verified.

State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x³ – x² – 6x + 2 in [0, 3]

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State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x3 – x2 – 6x + 2 in [0, 3]

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State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x³ – x² – 6x + 2 in [0, 3]