Question

Let $f:[0,2]\to R$ be a function defined by $f\left(x\right)=x^3-3x$, $x\in [0,2]$ then the complete interval where the function is increasing is

• A. $(1,2)$
• B. $[0,2)$
  
• C. $[0,1]$
• D. none of these

Answer 1

The correct option is A $(1,2)$

The function $f\left(x\right)$ is given by $f\left(x\right)=x^3-3x$, $x\in [0,2]$.

Now $f^{\prime }\left(x\right)=3x^2-3$.

If $f^{\prime }\left(x\right)>0$ then $3(x^2-1)>0$ or $(x^2-1)>0$.

Now $x^2-1>0$ gives $(x+1)(x-1)>0$.

$(x+1)(x-1)>0$ gives $x>-1$ and $x>1$.......(1) or $x<-1$ and $x<1$.

From (1) we get $x\in (1,2)$ is the interval where the function $f\left(x\right)$ is increasing.