In the temperature range between 0°C and 700°C the resistance R [in ohms] of a certain platinum resistance thermometer is given by R = 10 + 0.04124T − 1.779 × 10^(−5)T^2where T is the temperature in degrees Celsius. Where in the interval from 0°C to 700°C is the resistance of the thermometer most sensitive and least sensitive to temperature changes? [Hint: Consider the size of dR/dT in the interval 0 ≤ T ≤ 700.].

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As we know the resistance would be sensitive for we need to find:

dRdTcfrac{dR}{dT}dTdR​

so from here we find that :

ddT(10+0.04124T−1.779×10−5T2)cfrac{d}{dT}(10 + 0.04124T − 1.779 × 10^{−5}T^2)dTd​(10+0.04124T−1.779×10−5T2)

0.04124−2×1.779×10−5T0.04124-2times1.779times10^{-5}T0.04124−2×1.779×10−5T

Now it is clear that the dRdTcfrac{dR}{dT}dTdR​ is decreasing

Now the Resistance would be most sensitive when :

dRdTcfrac{dR}{dT}dTdR​ is 0,, so for here that would be

T=1159.07  oCT=1159.07 ^oCT=1159.07  oC

it would be least sensitive for :

dRdTcfrac{dR}{dT}dTdR​ is minimum

and for the given range it would be for :

T=700  oCT= 700 ^oCT=700  oC