Question:
Mark the correct alternative in the following:
In the interval $(1,2)$, function $f(x)=2|x-1|+3|x-2|$ is
A. monotonically increasing
B. monotonically decreasing
C. not monotonic
D. constant
Solution:
Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly decreading on $(a, b)$ is that $f^{\prime}(x)<0$ for all $x \in(a, b)$
Given:-
$f(x)=2(x-1)+3(2-x)$
$f(x)=-x+4$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=-1=\mathrm{f}(\mathrm{x})$
Therefore $f^{\prime}(x)<0$
Hence decreasing function