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

c such that

f ′ (c ) = 0, if

f (x ) = 1 - x ^{2} in the interval

[- 1, 1]. e f A.

f - 2

B.

f 0

C.

f 1 2 D.

f 1

E.

f - 1

f
B

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