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Attempt following question by selecting a choice to answer. D Use Rolle's Theorem to determine the value of

c such that

f ′ (c ) = 0, if

f (x ) = x ^{2} - 16x + 63 in the interval

[7, 9]. BB DD A.

D 8

B.

D 0

C.

D 9

D.

D 7

E.

D 1 8 D
A

Step 1: f (x ) = x ^{2} - 16x + 63[Write the function.] Step 2: f (7) = (7)^{2} - 16(7) + 63 = 0[Find f (7).] Step 3: f (9) = (9)^{2} - 16(9) + 63 = 0[Find f (9).] Step 4: Since f (x ) is continuous in [7, 9] and differentiable in (7, 9), also f (7) = f (9) = 0, Rolle's Theorem is applicable. Step 5: There exists a number c in (7, 9) such that f ′(c ) = 0. Step 6: f (x ) = x ^{2} - 16x + 63 ⇒ f ′(x ) = 2x - 16[Fine f ′ (x ).] Step 7: f ′(c ) = 2c - 16[Find f ′(c ).] Step 8: 2c - 16 = 0[Equate f ′(c ) to zero.] Step 9: c = 8 , which lies in the interval (7, 9).[Solve for c .] Step 10: So, the derivative of the function is zero at c = 8 in the interval [7, 9].