Answer
False.
Work Step by Step
Counterexample
Taking $f(x)=x\qquad g(x)=-x,$
see below
The area bounded by f(x) is $-0.5.$
The area bounded by g(x) is $-0.5$
The product, RHS=$-0.25=-\displaystyle \frac{1}{4}.$
The area bounded by $f(x)g(x)=-x^{2}$ , LHS,
is slightly greater in magnitude,$ -\displaystyle \frac{1}{3}$ .
$LHS=-\displaystyle \frac{1}{3},\qquad RHS=-\frac{1}{4}$
