Question

Let $f\left(x\right)=x^3-x^2+1$ be a continuous function in $R$.
Statement $1:f\left(x\right)$ has atleast one real solution in $(-2,2).$
Statement $2:f^{-1}\left(x\right)$ has atleast one real solution in $(-2,2).$
Then

• A. Statement $1$ is true, Statement $2$ is true and Statement $2$ is the correct explanation of Statement $1.$
• B. Statement $1$ is true, Statement $2$ is true and Statement $2$ is not the correct explanation of Statement $1.$
• C. Statement $1$ is true and Statement $2$ is false.
• D. Statement $1$ is false and Statement $2$ is true.

Answer 1

The correct option is C Statement $1$ is true and Statement $2$ is false.

Given : $f\left(x\right)=x^3-x^2+1$ is continuous in $[-2,2]$
$\Rightarrow f(-2)<0$ and $f\left(2\right)>0$
$\Rightarrow f\left(x\right)=0$ has atleast one solution in $(-2,2)$.
Also, $f\left(x\right)$ is not invertible as $f\left(0\right)=f\left(1\right)=1$ which is not one-one.