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Argue that, given your assumptions on the utility function, $x^*$ is the essentially unique (and hence global) maximum. (You need this because there may be local maxima when the assumptions on the utility function is relaxed - this will violate the proposition you're trying to prove). Now simply use the definition of global optima: for any $x\leq x^*$, $u(x)...


As correctly pointed out by @Herr K., the opportunity cost of forgoing a movie (bowling) and going bowling (to a movie) is $2. Hence, the net payoff from going to a movie = benefit - cost = 12 - (10+2) =0. Similarly, the net payoff from going bowling is 17-(15+2) = 0 Hence the consumer should be indifferent between bowling and going to a movie.

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