Question

the function g(x) is defined for 0 ≤ x ≤ 15. use the graph of g(x) to answer the questions below. when x = 1, what can you say about the concavity of the graph? concave up concave down neither concave up nor concave down when x = 8, what can you say about the concavity of the graph? concave up concave down neither concave up nor concave down when x = 14 what can you say about the concavity of the graph? concave up concave down neither concave up nor concave down write the open x - intervals for which the function is both increasing and concave up.

Explanation:

Step1: Recall concavity definition

Concave - up: curve opens upward, second - derivative positive. Concave - down: curve opens downward, second - derivative negative.

Step2: Analyze \(x = 1\)

At \(x = 1\), the curve is bending downward, so it is concave down.

Step3: Analyze \(x = 8\)

At \(x = 8\), the curve is bending upward, so it is concave up.

Step4: Analyze \(x = 14\)

At \(x = 14\), the curve is bending upward, so it is concave up.

Step5: Find increasing and concave - up intervals

The function is increasing when the slope is positive (going up from left to right) and concave up when it bends upward. The function is increasing and concave up on the interval \((9,15)\).

Answer:

When \(x = 1\), concave down.
When \(x = 8\), concave up.
When \(x = 14\), concave up.
The open \(x\) - intervals for which the function is both increasing and concave up is \((9,15)\).