Let f be a real-valued function continuous on the closed interval [a,b] and differentiable on (a,b).
a. (a) Define a critical point of a real-valued function.
b. (b) Give a clear reason why, when determining the absolute maximum or minimum value of f on [a,b], it is necessary to compare the values of f at the endpoints a,b with its values at all critical points in (a,b).
c. (c) A student asserts that if f′(c)=0 for some c∈(a,b), then f must attain its absolute maximum at c. Construct an explicit example of a function on [0,3] that invalidates this assertion, and justify your answer.