Let h be the function given by h(x)=x−3x2, defined for all real x=3. The domain is restricted to the closed interval [0,5].
(a) Determine all critical numbers of h that lie in the open interval (0,5) and show complete algebraic work.
(b) Classify each critical point from part (a) as a local maximum, local minimum, or neither. Provide analytic justification for every classification.
(c) Using the results from part (b) together with an analysis of the endpoints, decide the absolute maximum and absolute minimum values of h on [0,5]. State the x–values where these extrema occur and supply coherent reasoning.