Question

The domain of the function
$ f \left(x\right)=\sqrt{2x^{3}-5x^{2}-4x+10} $ is

• A. $ \left[-\sqrt{2}, \sqrt{2}\right] \cup [\frac{1}{2}, \infty) $
• B. $ \left[-\sqrt{3}, \sqrt{3}\right]\cup [\frac{5}{2}, \infty) $
• C. $ \left[-\sqrt{2}, \sqrt{2}\right]\cup [\frac{5}{2}, \infty) $
• D. $ \left[-\sqrt{2}, \sqrt{2}\right]\cup [\frac{5}{2}, 10) $

Answer 1

Answer: (C)

We have, $f(x) = \sqrt{2x^3 - 5x^2 - 4x +10}$
$f(x) $ is defined if $2x^3 - 5x^2 -4x + 10 \ge 0$
$\Rightarrow x^2(2x - 5) - 2(2x - 5) \ge 0$
$\Rightarrow (2x - 5)(x^2 -2) \ge 0$
$\Rightarrow x \ge \frac{5}{2}$ or $- \sqrt{2} \le x \le \sqrt {2}$
$\therefore D_f = [-\sqrt{2} , \sqrt{2} ] \cup [\frac{5}{2}, \infty)$