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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 ) = cos x in the interval

[- π 2 , π 2 ]. B D A.

D 0

B.

D π 4 C.

D π 2 D.

D -

π 2 E.

D π 6 D
A

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