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Consider the function f(x) = x^3-3x^2+2x-18 on the interval [ 0 , 2 ]. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval: f(x) is ___on[0,2] f(x) is ___ on (0,2) and f(0)=f(2)=____ then by rolle's theorem, there e...

Consider the function f(x) = 3-3 x^{2/3} on the interval [ -1 , 1 ]. Which of the three hypotheses of Rolle's Theorem fails for this function on the inverval? (a) f(x) is continuous on [-1,1]. (b) f(x) is differentiable on (-1,1). (c) f(-1)=f(1). Answer:(...

Consider the function f(x) = 3 -2 x^2 on the interval [ -5 , 3 ]. (A) Find the average or mean slope of the function on this interval, i.e. \displaystyle{\frac { f(3) - f(-5) }{ 3 - (-5) } = } 4 (B) By the Mean Value Theorem, we know there exists a c in t...

Consider the function f(x) = 1/x on the interval [ 4 , 11 ]. (A) Find the average or mean slope of the function on this interval. Average Slope = (B) By the Mean Value Theorem, we know there exists a c in the open interval ( 4 , 11 ) such that f'( c) is e...

Consider the function f(x) = 1/x on the interval [ 4 , 11 ]. (A) Find the average or mean slope of the function on this interval. Average Slope =_______ (B) By the Mean Value Theorem, we know there exists a c in the open interval ( 4 , 11 ) such that f'(...