Let f be a continuous function on [a,b] and let k∈R. Define h(x)=f(x)+k on the same interval.
a. (i) State the definition of the average value of a continuous function on a closed interval.
b. (ii) Prove that the average value of h on [a,b] is equal to the average value of f on [a,b] increased by k.
c. (iii) Take f(x)=x2 on [0,3] and k=−5. Find the average value of h(x)=x2−5 on [0,3] and verify that your result is consistent with part (ii).