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How to prove mathematically that the sliding rheostat has the largest resistance when the slider is centered? Please answer in many ways, preferably including the mean value theorem.
You should be talking about the total resistance of the two parts of the sliding rheostat in parallel.

Let the resistances of the two parts be R 1 and R2, and the total resistance of the resistance wire of the sliding rheostat be R, then there are

R 1+R2=R

Total resistance of parallel circuit

R joint =R 1R2/(R 1+R2)

R 1+R2≥2√(R 1R2). Since R 1+R2 is a constant value, when R 1=R2, R1R2 takes the maximum value. At this time,

R joint =R/2

That is, when the slider is in the middle, the resistance is the greatest.

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