Question

the following is the graph of a function f = f(x). where is the graph decreasing and concave up? (select all that apply.) 0 < x < 0.5 0.5 < x < 1 1 < x < 1.5 1.5 < x < 2 2 < x < 2.5 2.5 < x < 3 3 < x < 3.5 3.5 < x < 4

Explanation:

Step1: Recall function properties

A function is decreasing when $f'(x)<0$ and concave - up when $f''(x)>0$. Visually, decreasing means the graph goes down as $x$ increases, and concave - up means the graph curves upward like a cup.

Step2: Analyze the intervals

• For $0 < x<0.5$: The graph is decreasing but concave - down.
• For $0.5 < x<1$: The graph is increasing.
• For $1 < x<1.5$: The graph is decreasing and concave - up.
• For $1.5 < x<2$: The graph is decreasing and concave - down.
• For $2 < x<2.5$: The graph is increasing.
• For $2.5 < x<3$: The graph is decreasing and concave - up.
• For $3 < x<3.5$: The graph is decreasing and concave - down.
• For $3.5 < x<4$: The graph is increasing.

Answer:

$1 < x<1.5$, $2.5 < x<3$