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The range of values of k for which the function $f (x) = (k^{2} - 7 k + 12) \cos x + 2 (k - 4) x + \log 2$ does not possess critical points is

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$(1, 5) - {4}$

(1,7)

(1,5)

(1,4)

answer is B.

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Detailed Solution

$\begin{array}{l} F^{1} (x) = (k^{2} - 7 k + 12) (- \sin x) + 2 (k - 4) \\ = (k - 4) (k - 3) (- \sin x) + 2 (k - 4) \\ = (k - 4) [- (k - 3) \sin x + 2) \neq 0 k - 4 \neq 0 and \sin x \neq \frac{2}{k - 3} \end{array}$

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