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Find the point c in the curve f(x) = x3 + x2 + x + 1 in the interval where slope of a tangent to a curve is equals to the slope of a line joining (0,1)

Find the point c in the curve f(x) = x3 + x2 + x + 1 in the interval where slope of a tangent to a curve is equals to the slope of a line joining (0,1) Correct Answer 0, 1

f(x) = x3 + x2 + x + 1 f(x) is continuous in given interval . f’(x) = 3x2+2x+1 Since, value of f’(x) is always finite in interval (0, 1) it is differentiable in interval (0, 1). f(0) = 1 f(1) = 4 By mean value theorem, f’(c) = 3c2 + 2c + 1 = (4-1)/(1-0) = 3 ⇒ c = 0.548,-1.215 Since c belongs to (0, 1) c = 0.54.

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