Question

The range of \(a\in\mathbb R\) for which the function \(f(x)=(4a-3)(x+\log_e5)+2(a-7)\cot\left(\frac{x}{2}\right)\sin^2\left(\frac{x}{2}\right)\), \(x\ne2n\pi,\ n\in N\) has critical points, is :

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

Hint:

Simplify \(\cot\frac x2\sin^2\frac x2=\frac12\sin x\), so \(f'(x)=(4a-3)+(a-7)\cos x\); critical points exist iff \(\cos x=\frac{3-4a}{a-7}\) is solvable.

Answer 1

The Correct Option is C

\(\cot\frac x2\sin^2\frac x2=\sin\frac x2\cos\frac x2=\frac12\sin x\), so \(f(x)=(4a-3)(x+\log_e5)+(a-7)\sin x\).
\(f'(x)=(4a-3)+(a-7)\cos x\).
Critical points need \(f'(x)=0\). If \(a=7\), \(f'=25\ne0\): none. Otherwise \(\cos x=\frac{3-4a}{a-7}\), requiring \(\left|\frac{3-4a}{a-7}\right|\le1\).
\((3-4a)^2\le(a-7)^2\Rightarrow15a^2-10a-40\le0\Rightarrow3a^2-2a-8\le0\Rightarrow(3a+4)(a-2)\le0\Rightarrow a\in\left[-\frac43,2\right]\).
Answer: (3) \(\left[-\frac{4}{3},2\right]\)