Question

If $f:[2,3]\to R$ 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]

Answer 1

Answer: (B)

Given, $f(x)=x^3+3x-2$
On differentiating w.r.t. $x$, we get
$f^{\prime }(x)=3x^2+3$
Put $f^{\prime }(x)=0\Rightarrow 3x^2+3=0$
$\Rightarrow x^2=-1$
$\therefore f(x)$ is either increasing or decreasing.
At $x=2,f(2)=2^3+3(2)-2=12$
At $x=3,f(3)=3^3+3(3)-2=34$
$\therefore f(x)\in [12,34]$