Q. If $f:\lbrack 2,\ 3\rbrack \rightarrow R$ is defined by $f(x) = x^{3} + 3x - 2$, then the range $f(x)$ is contained in the interval
A) [1, 12]
B) [12, 34]
C) [35, 50]
D) [-12, 12]
Explanation
(b)
Given, $f(x) = x^{3} + 3x - 2$
On differentiating w.r.t. $x$, we get
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f^{'}(x) = 3x^{2} + 3$
Put $f^{'}(x) = 0\ \Rightarrow 3x^{2} + 3 = 0$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^{2} = - 1$
$\therefore\ \ f(x)$ is either increasing or decreasing.
At $x = 2,\ f(2) = 2^{3} + 3(2) - 2 = 12$
At $x = 3,\ f(3) = 3^{3} + 3(3) - 2 = 34$
$\therefore f(x) \in \lbrack 12,\ 34\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.