Question

The value of $a$ for which the function $f\left(x\right)=(4a-3)(x+\log 5)+2(a-7)\cot \frac{x}{2}{\sin }^2\frac{x}{2}$ does not possess critical points is

• A. $(-\infty ,-\frac{4}{3})$
• B. $(-\infty ,-1)$
• C. $[1,\infty )$
• D. $(2,\infty )$

Answer 1

Answer: (A), (D)

$(-\infty ,-\frac{4}{3})$
$(2,\infty )$

$f\left(x\right)=(4a-3)(x+\log 5)+2(a-7)\cot \left(x/2\right){\sin }^{2}x/2f\left(x\right)=(4a-3)(x+\log 5)+(a-7)(2\times \cot (x/2\left){\sin }^{2}x/2\right)f\left(x\right)=(4a-3)(x+\log 5)+(a-7)\left(\sin x\right)f^{\prime }\left(x\right)=(4a-3)+(a-7)\left(\cos x\right)$

$f^{\prime }\left(x\right)$ will not be zero

if $(4a-3)>(a-7)$

$2a>-4$

$a>-2$

or

$4a-3<-(a-7)$

$5a<10$

$a<2$