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

f (x ) = x 2 - 9 on

[3, 9]. D D A.

D 3

2 B.

D 0

C.

D - 3

3 D.

D 3

E.

D 3

3 D
A

Step 1: f (x ) = x 2 - 9 [Write the function.] Step 2: Since f (x ) is continuous in [3, 9] and differentiable in (3, 9), Mean Value Theorem is applicable. Step 3: f (3) = ( 3 ) 2 - 9 = 0[Find f (3).] Step 4: f (9) = ( 9 ) 2 - 9 = 62 [Find f (9).] Step 5: f ′(x ) = x x 2 - 9 [Find f ′(x ).] Step 6: f ′(c ) = c c 2 - 9
[Find f ′(c ).] Step 7: By Mean Value Theorem, there exists c ∈ (3, 9) such that f ′(c ) = f ( 9 ) - f ( 3 ) 9 - 3 . Step 8: c c 2 - 9 = 6 2 - 0 6 Step 9: c c 2 - 9 = 2 Step 10: c = 32 ∈ (3, 9)[Solve for c .] Step 11: At c = 32 in (3, 9), the function f (x ) satisfies the Mean Value Theorem.