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Attempt following question by selecting a choice to answer. f Find the value in the interval, which satisfies the Mean Value Theorem for the function

f (x ) = 4x ^{3} - 5x ^{2} + x - 2 on

[0, 1]. c f A.

f 5 + 1 3 4 B.

f 5 + 1 3 6 C.

f 5 - 1 3 1 2 D.

f 1 4 E.

f - 2

f
C

Step 1: f (x ) = 4x ^{3} - 5x ^{2} + x - 2[Write the function.] Step 2: Since f (x ) is continuous in [0, 1] and differentiable in (0, 1), Mean Value Theorem is applicable. Step 3: f (0) = 4(0)^{3} - 5(0)^{2} + 0 - 2 = - 2[Find f (0).] Step 4: f (1) = 4(1)^{3} - 5(1)^{2} + 1 - 2 = - 2[Find f (1).] Step 5: f ′(x ) = 12x ^{2} - 10x + 1[Find f ′(x ).] Step 6: f ′(c ) = 12c ^{2} - 10c + 1[Find f ′(c ).] Step 7: By Mean Value Theorem, there exists c ∈ (0, 1) such that f ′(c ) = f ( 1 ) - f ( 0 ) 1 - ( 0 ) . Step 8: 12c ^{2} - 10c + 1 = - 2 - ( - 2 ) 1 - 0
Step 9: 12c ^{2} - 10c + 1 = 0 Step 10: c = 5 - 1 3 1 2 , 5 + 1 3 1 2 ∈ (0, 1)
[Use quadratic formula.] Step 11: At x = 5 + 1 3 1 2 , 5 - 1 3 1 2 in (0, 1), the function f (x ) satisfies the Mean Value Theorem.