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Chapter 3: Q. 1 (page 286)

Local and global extrema:

Use mathematical notation, including inequalities as used in the definition of local and global extrema, to express each of the following statements.

On the interval [−3, 5], fhas a local maximum at x = 2.
On the interval [0, 2], f has a global maximum at x = −1.

On the interval [−4, 4], f has a global minimum at x = 0.
On the interval [0, 5], fhas no global minimum.

Short Answer

Expert verified

f has a local maximum at x=2 if f(2)>f(x) for all x near x=2.

f has no global minimum and the graph does not have the lowest point in the domain of f.

Step by step solution

01

Step 1. Given Information.

On the interval [−3, 5], fhas a local maximum at x = 2.
On the interval [0, 2], f has a global maximum at x = −1.

On the interval [−4, 4], f has a global minimum at x = 0.
On the interval [0, 5], fhas no global minimum.

02

Step 2. Write it in mathematical notation.

The function f has some local maximum at x=2if there exists δ>0such that f(2)≥f(x)for all x∈(2-δ,2+δ)for all near x=2.

f has a global maximum at x=2 if f(2)≥f(x)forallx∈Domain of f.

f has a global minimum at x=0 if f(0)≤f(x)forallx∈ Domain of f .

f has no global minimum and the graph does not have the lowest point in the domain of f.

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