The range of a[]R for which the function f(x) = (4a - 3)(x + loge 5) + 2(a - 7) cot[( x 2 …

Question

The range of a$$\in$$R for which the

function f(x) = (4a - 3)(x + loge 5) + 2(a - 7) cot$$\left( {{x \over 2}} \right)$$ sin2$$\left( {{x \over 2}} \right)$$, x $$\ne$$ 2n$$\pi$$, n$$\in$$N has critical points, is :

Accepted Answer

$$f(x) = (4a - 3)(x + \ln 5) + 2(a - 7)\left( {{{\cos {x \over 2}} \over {\sin {x \over 2}}}.{{\sin }^2}{x \over 2}} \right)$$

$$f(x) = (4a - 3)(x + \ln 5) + (a - 7)\sin x$$

$$f'(x) = (4a - 3) + (a - 7)\cos x = 0$$

$$\cos x = {{ - (4a - 3)} \over {a - 7}}$$

$$ - 1 \le  - {{4a - 3} \over {a - 7}} \le 1$$

$$ - 1 \le {{4a - 3} \over {a - 7}} \le 1$$

$${{4a - 3} \over {a - 7}} - 1 \le 0$$ and $${{4a - 3} \over {a - 7}} + 1 \ge 0$$

$$ \Rightarrow {{ - 4} \over 3} \le a \le 2$$

Solutions · 1