Mark the correct alternative in the following:
Question:

Mark the correct alternative in the following:

In the interval $(1,2)$, function $f(x)=2|x-1|+3|x-2|$ is

A. increasing

B. decreasing

C. constant

D. none of these

Solution:

Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly decreasing on $(a, b)$ is that $f^{\prime}(x)<0$ for all $x \in(a, b)$

Given:-

$f(x)=2(x-1)+3(2-x)$

$\Rightarrow f(x)=-x+4$

$\frac{d(f(x))}{d x}=f^{\prime}(x)=-1$

Therefore $f^{\prime}(x)<0$

Hence decreasing function