Answer
$t=1.3\hspace{0.1cm}\text{hrs}$
Work Step by Step
The thermic effect of food is given by,
$F(t)=-10.28+175.9te^{\frac{-t}{1.3}}$
To find the maximum time we first find the critical points of $F(t)$.
$F'(t)=0\implies 175.9e^{\frac{-t}{1.3}}(1-\frac{t}{1.3})=0$
$\implies t=1.3\hspace{0.1cm}\text{hrs}$
To verify if $F(t)$ is maximum at $t=1.3$, check the sign of derivative at both the sides of $t=1.3$.
$F'(1)=175.9e^{\frac{−1}{1.3}}(1−\frac{1}{1.3})>0$
$F'(2)=175.9e^{\frac{−2}{1.3}}(1−\frac{2}{1.3})<0$
Since the sign changes from $+$ to $-$, by the first derivative test, the thermic effect of food is maximum at $t=1.3\hspace{0.1cm}\text{hrs}$
