Question

If $x=1$ is the point of extrema for $f\left(x\right)=\{\begin{array}{cc}x,& 0<x<1\\ g\left(x\right),& 1\le x<2\end{array},$ then which of the following(s) is(are) correct

• A. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)<1$
• B. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)<1$
• C. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)>1$
• D. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)>1$

Answer 1

Answer: (A), (C)

$g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)>1$

Given : $f\left(x\right)=\{\begin{array}{cc}x,& 0<x<1\\ g\left(x\right),& 1\le x<2\end{array}$

Case $\left(A\right):$ If $x=1$ is a point of minima
$\Rightarrow f(1-h)>f\left(1\right)<f(1+h),\text{where}h\to 0^{+}$
$\Rightarrow 1>g\left(1\right)<g(1+h),h\to 0^{+}$
$\Rightarrow g\left(1\right)<1$ and $g\left(1\right)<g(1+h)\left(\text{i.e. increasing at}x=1\right)$
So, $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)<1$

Case $\left(B\right):$ If $x=1$ is a point of maxima
$\Rightarrow f(1-h)<f\left(1\right)>f(1+h),\text{where}h\to 0^{+}$
$\Rightarrow 1<g\left(1\right)>g(1+h),h\to 0^{+}$
$\Rightarrow g\left(1\right)>1$ and $g\left(1\right)>g(1+h)\left(\text{i.e. decreasing at}x=1\right)$
So, $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)>1$