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R

V_{m a x} = \frac{R}{R + R 0} V_{i n}

V_{m i n} = \frac{R}{R + R 1} V_{i n}

V_{d i f f} = (\frac{R}{R + R 0} - \frac{R}{R + R 1}) V_{i n}

\frac{V_{d i f f}}{V_{i n}} = (\frac{R}{R + R 0} - \frac{R}{R + R 1})

\frac{d \frac{V_{d i f f}}{V_{i n}}}{d R} = (\frac{1}{R + R 0} - \frac{R}{{(R + R 0)}^{2}} - \frac{1}{R + R 1} + \frac{R}{{(R + R 1)}^{2}})

= (\frac{1}{R + R 0} - \frac{1}{R + R 1}) + (\frac{R}{{(R + R 1)}^{2}} - \frac{R}{{(R + R 0)}^{2}})

= (\frac{(R + R 0)}{{(R + R 0)}^{2}} - \frac{(R + R 1)}{{(R + R 1)}^{2}}) + (\frac{R}{{(R + R 1)}^{2}} - \frac{R}{{(R + R 0)}^{2}})

= (\frac{R 0}{{(R + R 0)}^{2}} - \frac{R 1}{{(R + R 1)}^{2}})

= \frac{R 0 {(R + R 1)}^{2} - R 1 {(R + R 0)}^{2}}{{(R + R 1)}^{2} {(R + R 0)}^{2}}

R 0 {(R + R 1)}^{2} - R 1 {(R + R 0)}^{2} = 0

R 0 R^{2} + 2 R R 0 R 1 + R 0 R 1^{2} - R 1 R^{2} - 2 R R 0 R 1 - R 1 R 0^{2} = 0

R 0 R^{2} - R 1 R^{2} = R 1 R 0^{2} - R 0 R 1^{2} \Rightarrow R^{2} = \frac{R 1 R 0^{2} - R 0 R 1^{2}}{R 0 - R 1} = R 0 R 1

R = \sqrt{R 0 \times R 1}

V_{m a x} = \frac{R 1}{R + R 1} V_{i n}

V_{m i n} = \frac{R 0}{R + R 0} V_{i n}

V_{d i f f} = (\frac{R 1}{R + R 1} - \frac{R 0}{R + R 0}) V_{i n}

\frac{V_{d i f f}}{V_{i n}} = (\frac{R 1}{R + R 1} - \frac{R 0}{R + R 0})

\frac{d \frac{V_{d i f f}}{V_{i n}}}{d R} = (- \frac{R 1}{{(R + R 1)}^{2}} + \frac{R 0}{{(R + R 0)}^{2}}) = 0

R 0 {(R + R 1)}^{2} - R 1 {(R + R 0)}^{2} = 0

R = \sqrt{R_{m i n} \times R_{m a x}}

Maximising the range of a resistive sensor.