# Answer on Calculus Question for hsd

Question #17231

Which of the following statements are true (T) and

which are false (F)? Justify if true and give a counter-example if false.

Suppose that f(x) has domain [1; 3] and is continuous, that g(x) has domain R and is

dierentiable, and that h(x) has domain (1; 3) and is differentiable.

(a) There is some a E R with 1 <= a <= 3 such that f has a global maximum at a.

(b) There is some a E R with 1 < a < 3 such that f has a global maximum at a.

(c) There is some a E R with 1 < a < 3 such that f has a local minimum at a.

(d) f cannot have infinitely many different local maximum points.

(e) There is some a E R such that g has a global maximum at a.

(f) g has at least one local extremum.

(g) If h has a local maximum at a 2 (1; 3), then h'(a) = 0.

(h) If h'(a) = 0 for some a E (1; 3), then h has a local maximum or a local minimum at a.

which are false (F)? Justify if true and give a counter-example if false.

Suppose that f(x) has domain [1; 3] and is continuous, that g(x) has domain R and is

dierentiable, and that h(x) has domain (1; 3) and is differentiable.

(a) There is some a E R with 1 <= a <= 3 such that f has a global maximum at a.

(b) There is some a E R with 1 < a < 3 such that f has a global maximum at a.

(c) There is some a E R with 1 < a < 3 such that f has a local minimum at a.

(d) f cannot have infinitely many different local maximum points.

(e) There is some a E R such that g has a global maximum at a.

(f) g has at least one local extremum.

(g) If h has a local maximum at a 2 (1; 3), then h'(a) = 0.

(h) If h'(a) = 0 for some a E (1; 3), then h has a local maximum or a local minimum at a.

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