the graph of a function is given. (a) find all the local maximum and minimum values of the function as well…

the graph of a function is given. (a) find all the local maximum and minimum values of the function as well as the value of x at which each occurs. local maximum (x, y) = ( 0,4 ) × local minimum (x, y) = ( -3,1 ) (smaller x - value) × local minimum (x, y) = ( 2,2 ) (larger x - value) (b) find the intervals on which the function is increasing, and on which the function is decreasing. increasing: -3, 0 ∪ 2, ∞) (-∞, -3 ∪ 0, 2 -4, 8 ∪ 2, ∞) 0, ∞) decreasing: -3, 2 (-∞, 0 (-∞, -3 ∪ 0, 2 -3, 0 ∪ 2, ∞) (∞, -4 ∪ 8, 2

the graph of a function is given. (a) find all the local maximum and minimum values of the function as well as the value of x at which each occurs. local maximum (x, y) = ( 0,4 ) × local minimum (x, y) = ( -3,1 ) (smaller x - value) × local minimum (x, y) = ( 2,2 ) (larger x - value) (b) find the intervals on which the function is increasing, and on which the function is decreasing. increasing: -3, 0 ∪ 2, ∞) (-∞, -3 ∪ 0, 2 -4, 8 ∪ 2, ∞) 0, ∞) decreasing: -3, 2 (-∞, 0 (-∞, -3 ∪ 0, 2 -3, 0 ∪ 2, ∞) (∞, -4 ∪ 8, 2

Answer

Explanation:

Step1: Identify local extrema

Local maximum occurs where the function changes from increasing to decreasing. Local minimum occurs where the function changes from decreasing to increasing. By observing the graph, the local maximum is at the point where the curve reaches a peak. The local minima are at the points where the curve reaches valleys.

Step2: Determine increasing intervals

A function is increasing when the y - values increase as the x - values increase. Looking at the graph, we see the function is increasing when (x\in[-3,0]\cup[2,\infty)).

Step3: Determine decreasing intervals

A function is decreasing when the y - values decrease as the x - values increase. From the graph, the function is decreasing when (x\in(-\infty,-3]\cup[0,2]).

Answer:

(a) local maximum ((x,y)=(0, 4)) local minimum ((x,y)=(-3,1)) (smaller (x) - value) local minimum ((x,y)=(2,2)) (larger (x) - value) (b) Increasing: ([-3,0]\cup[2,\infty)) Decreasing: ((-\infty,-3]\cup[0,2])