Sensitivity of a thermocouple from thermometric function

[Kavorka]

Homework Statement

In what range of temperature will this thermocouple be more sensitive (i.e. having a measureable voltage for a small change in temperature)?

Homework Equations

The thermal electromotive force (E) of a thermocouple is described in terms of temperature by the function:
E = (1/2)T - (1/1000)T² with E in mV, T in degrees Celsius
For previous questions, I have graphed the function and found its maximum. They want the answer in terms of the graph, and although I have an idea what the answer is I'm not 100% sure that I am right.

The Attempt at a Solution

My answer:

The greatest thermocouple sensitivity or greatest ΔmV for the smallest ΔT will be when |dE| is largest, or when the temperature is farthest away from the dE=0 extrema at T=250°C. AT the farthest temperatures from T=250°C within the operational limits of the thermocoupler wherein f(E) still holds (too hot will melt the metal and reduce conductivity, too cold will limit conductivity) are where the smallest change in T results in the greatest measurable change in voltage. As we want the thermocouple to create a positive volatage, the most sensitive thermocouple temperatues where E is positive are close to the roots of f(E) at 0°C and 500°C.

My problem is I'm not sure if they mean the thermocouple is most sensitive only when the slope is large or when it its absolute value is large, or what exactly I should say about the limits of the function at extreme temperatures when the theoretical graph indicates the highest sensitivity but where in reality the device wouldn't function. Any tips would be helpful!

 

[andrevdh]

The greatest sensitivity should be where the gradient is the steepest. That is for a small change in temperature the output voltage changes the most. That means around the region(s) where the derivative of the function have extreme values.