Q. 1 - Calculus

Question

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], f has 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], f has no global minimum.

Step-by-Step Solution

Answer

f has a local maximum at x=2 if $f\left(2\right)>f\left(x\right)$ 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 1. Given Information.

On the interval [−3, 5], f has 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], f has no global minimum.

Step 2. Write it in mathematical notation.

The function f has some local maximum at $x=2$ if there exists $\delta >0$ such that $f\left(2\right)\ge f\left(x\right)$ for all $x\in (2-\delta ,2+\delta )$ for all near $x=2$.

f has a global maximum at x=2 if $f\left(2\right)\ge f\left(x\right) for all x\in$Domain of f.

f has a global minimum at x=0 if $f\left(0\right)\le f\left(x\right) for all x\in$ Domain of f .

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