If $f:\left[2,3\right]\to R$ is defined as $f\left(x\right)=x^3+3x-2$ then the range of the function $f\left(x\right)$ contained in the interval 

The given funciton  $f:\left[2,3\right]\to R$ is defined as $f\left(x\right)=x^3+3x-2$

The derivative of the fucntion is $f\text{'}\left(x\right)=3x^2+3$

This is always positive, it means the function $f\left(x\right)=x^3+3x-2$ is increasing funciton

Hence the range of the funciton is $\left[f\left(2\right),f\left(3\right)\right]$

It implies that the range of the function is $\left[12,34\right]$