Question

Let the function $f\left(x\right)$ be defined as below
$f\left(x\right)=\sin {}^{-1}\lambda +x^2,0<x<1$
$2x,x\ge 1$
$f\left(x\right)$can have a minimum at $x=1$ if the value of $\lambda$ is

• A. $1$
• B. $-1$
• C. $0$
• D. none of these

Answer 1

Answer: (D)

none of these

$f\left(x\right)=\{\begin{array}{c}{\sin }^{-1}\lambda +x^2,0<x<1\\ 2x,x\ge 1\end{array}$

${\sin }^{-1}\lambda +x^2$ is an increasing function and it's minimum value is at $x=0$, which is ${\sin }^{-1}\lambda$

Thus for f to have minima at $x=1$,

$2\left(1\right)<{\sin }^{-1}\lambda \Rightarrow {\sin }^{-1}\lambda >2,$ which is not possible since ${\sin }^{-1}\lambda \in [-\frac{\pi }{2},\frac{\pi }{2}]<2$