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

f (x ) = 1 + 1 x on

[1, 4]. c c A.

c - 2

B.

c 0

C.

c 2

D.

c 3 9 6 E.

c 3 5 1 0 c
C

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