u/Shot-Deal7786

Can someone please explain how to do these questions in details? Really appreciate it!

Hi guys!
Could someone please help me with this short answer question, especially part e and part f.
I have looked at the sample answers to these questions. For part e, the answer said that we just need to set dT/dx=0 at x=1 and then solve for k. But my question is, by just setting the derivative equal to 0 at x=1, how do you ensure it is a minimum, why can’t it be a maximum or something else like a point of inflection?
For part f, again the sample answer said that we only need to set dT/dx to be less than or equal to 0 at x=root 7/2, and solve the inequality for k.
My question is how do you know by setting an inequality like that can guarantee there is a minimum?
I am so confused and really want to know the techniques or strategies to do this type of questions.
Thank you!