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

f (x ) = x + 2 2 x on

[1 2 , 2]. ff BB A.

B 0

B.

B 1

C.

B 5 2 D.

B 7 3 E.

B - 1

B
B

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