Answer:
a)
The combined resistance of a circuit consisting of two resistors in parallel is given by:
[tex]\frac{1}{R}=\frac{1}{r_1}+\frac{1}{r_2}[/tex]
where
R is the combined resistance
[tex]r_1, r_2[/tex] are the two resistors
We can re-write the expression as follows:
[tex]\frac{1}{R}=\frac{r_1+r_2}{r_1r_2}[/tex]
Or
[tex]R=\frac{r_1 r_2}{r_1+r_2}[/tex]
In order to see if the function is increasing in r1, we calculate the derivative with respect to r1: if the derivative if > 0, then the function is increasing.
The derivative of R with respect to r1 is:
[tex]\frac{dR}{dr_1}=\frac{r_2(r_1+r_2)-1(r_1r_2)}{(r_1+r_2)^2}=\frac{r_2^2}{(r_1+r_2)^2}[/tex]
We notice that the derivative is a fraction of two squared terms: therefore, both factors are positive, so the derivative is always positive, and this means that R is an increasing function of r1.
b)
To solve this part, we use again the expression for R written in part a:
[tex]R=\frac{r_1 r_2}{r_1+r_2}[/tex]
We start by noticing that there is a limit on the allowed values for r1: in fact, r1 must be strictly positive,
[tex]r_1>0[/tex]
So the interval of allowed values for r1 is
[tex]0<r_1 <+\infty[/tex]
From part a), we also said that the function is increasing versus r1 over the whole domain. This means that if we consider a certain interval
a ≤ r1 ≤ b
The maximum of the function (R) will occur at the maximum value of r1 in this interval: so, at
[tex]r_1=b[/tex]
