Q. 8 - Calculus

Question

Given the following graph of f , graphically estimate the global extrema of f on each of the six intervals listed: 

$$\begin{gathered} \left(a\right) \left[-2,4\right] \left(b\right) \left(-2,4\right) \left(c\right) \left(-1,1\right) \\ \left(d\right) (0,4] \left(e\right) [0,4) \left(f\right) \left(-\infty ,\infty \right) \end{gathered}$$

Step-by-Step Solution

Answer

Part (a) Minimum = -2        Maximum = 4

Part (b) Minimum =  2     Maximum = 0

Part (c) Maximum = 0

Part (d) Minimum = 2        Maximum = 0

Part (e) Minimum = 2        Maximum = 4

Part (f) Minimum = -2,2        Maximum = 0,4

Part (a) Step 1. Given information.

Given graph is :

We have to graphically estimate the global extrema of f on:

$$\begin{gathered} \left(a\right) \left[-2,4\right] \left(b\right) \left(-2,4\right) \left(c\right) \left(-1,1\right) \\ \left(d\right) (0,4] \left(e\right) [0,4) \left(f\right) \left(-\infty ,\infty \right) \end{gathered}$$

Part (a) Step 2. Global extrema of f on [-2,4].

It is seen that the function f has a global minimum at x=-2 as the function is decreased up to that point and the function f has a global maximum at x=4 as the function is increased to that point.

Part (b) Step 1. Global extrema of f on (-2,4).

It is seen that in between the (-2,4), the function f has a global minimum at x=2 as the function is decreases up to that point and the function f has a global maximum at x=0 as the function is increases to that point.

Part (c) Step 1. Global extrema of f on (-1,1).

It is seen that in between the (-1,1), the function f has does not have a global minimum and the function f has a global maximum atx=0 as the function is increases to that point.

Part (d) Step 1. Global extrema of f on (0,4].

It is seen that in between the (0,4], the function f has a global minimum at function is decreases up to that point and the function f has a global maximum at x=0 as the function is increases to that point.

Part (e) Step 1. Global extrema of f on [0,4).

It is seen that in between the [0,4), the function f has a global minimum at x=2 as the function is decreases up to that point and the function f has a global maximum at x=4 as the the function is increases to that point.

Part (f) Step 1. Global extrema of f on - ∞ , ∞ .

It is seen that in between the $\left(-\infty ,\infty \right)$, the function f has a global minimum at x=-2 and x = 2 as the function is decreases up to that point and the function f has a global maximum at x = 0 and x=4 as the function is increases to that point.