Question

If $f:[2,3] \rightarrow R$ is defined by $f(x)=x^{3}+3 x-2$, then the range $f(x)$ is contained in the interval

• A. [1,12]
• B. [12,34]
• C. [35,50]
• D. [-12,12]

Answer 1

Answer: (B)

Given, $f(x)=x^{3}+3 x-2$
On differentiating w.r.t. $x$, we get
$f'(x)=3 x^{2}+3$
Put $f'(x)=0 \Rightarrow 3 x^{2}+3=0$
$\Rightarrow x^{2}=-1$
$\therefore f(x)$ is either increasing or decreasing.
At $x=2,\, f(2)=2^{3}+3(2)-2=12$
At $x=3,\, f(3)=3^{3}+3(3) -2=34$
$\therefore f(x) \in[12,34]$