For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

Concave down: $$\left$(0,$\frac{\pi}{2}\]\right$)$ ; Concave up: $$\left$($\frac{\pi}{2}$,$\pi\]\right$)$ ; Inflection pt: $$\left$($\frac{\pi}{2}$,0$\right$)$

Concave down: $$\left$(0,$\frac{\pi}{2}\]\right$)$ ; Concave up: $$\left$($\frac{\pi}{2}$,$\pi\]\right$)$ ; Inflection pt: $$\left$(0,$\frac{\pi}{2}\]\right$)$

Concave down: $$\left$(0,$\frac{\pi}{2}\]\right$)$ ; Concave up: $$\left$($\frac{\pi}{2}$,$\pi\]\right$)$ ; Inflection pts: $$\left$($\frac{\pi}{4}$,5$\right$)$ , $$\left$($\frac{3\pi}{4}$,-5$\right$)$

Concave down: $$\left$($\frac{\pi}{4}$,$\frac{3\pi}{4}\]\right$)$ ; Concave up: $$\left$(0,$\frac{\pi}{4}\]\right$)$ , $$\left$($\frac{3\pi}{4}$,$\pi\]\right$)$ ; Inflection pts: $$\left$($\frac{\pi}{4}$,5$\right$)$ , $$\left$($\frac{3\pi}{4}$,-5$\right$)$

Verified step by step guidance

To determine where the function is concave up or down, we need to analyze the second derivative of the function. Concavity is determined by the sign of the second derivative: if it's positive, the function is concave up; if it's negative, the function is concave down.

Identify the points where the concavity changes, known as inflection points. These occur where the second derivative is zero or undefined, and the sign of the second derivative changes.

From the graph, observe the shape of the curve. The function appears to be concave down from x = 0 to x = π/2, as the curve is bending downwards.

The function appears to be concave up from x = π/2 to x = π, as the curve is bending upwards.

The inflection point, where the concavity changes, is at x = π/2. At this point, the curve changes from concave down to concave up.

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