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Problem 3. Determine if each of the following sentences is TRUE or FALSE. If true, give a reason (for instance, a derivative test). If false, draw an example of a graph that is a counterexample to the sentence. For each of the sentences, assume that f(x) is twice-differentiable on (-infinity, infinity); this means that both f?(c) and f??(c) exist for all real numbers c. (a) (True/False) If f'(2) = 8, then f (x) has a local minimum at x = 2. (b) (True/False) If f?(2) = 0, then f (x) has a local minimum at x = 2. (c) (True/False) If f'(2) = 0 and f??(2) = 8, then f (x) has a local minimum at x = 2. (d) (True/False) If f'(2) = 0 and f(x) switches from increasing to decreasing at x = 2, then f(x) has a local minimum at x = 2. (e) (True/False) If f'(2) = 0 and f' (x) switches from increasing to decreasing at x = 2, then, f (x) has a local minimum at x = 2.
Show transcribed image text Problem 3. Determine if each of the following sentences is TRUE or FALSE. If true, give a reason (for instance, a derivative test). If false, draw an example of a graph that is a counterexample to the sentence. For each of the sentences, assume that f(x) is twice-differentiable on (-infinity, infinity); this means that both f?(c) and f??(c) exist for all real numbers c. (a) (True/False) If f'(2) = 8, then f (x) has a local minimum at x = 2. (b) (True/False) If f?(2) = 0, then f (x) has a local minimum at x = 2. (c) (True/False) If f'(2) = 0 and f??(2) = 8, then f (x) has a local minimum at x = 2. (d) (True/False) If f'(2) = 0 and f(x) switches from increasing to decreasing at x = 2, then f(x) has a local minimum at x = 2. (e) (True/False) If f'(2) = 0 and f' (x) switches from increasing to decreasing at x = 2, then, f (x) has a local minimum at x = 2.