Question

question 7
the function graphed above is:
concave up on the interval(s)
concave down on the interval(s)
there is an inflection point at:
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Explanation:

Step1: Recall concavity definitions

A function \(y = f(x)\) is concave up when \(f''(x)>0\) (the graph lies above its tangent lines) and concave down when \(f''(x)<0\) (the graph lies below its tangent lines). An inflection point is a point where the concavity changes, i.e., \(f''(x)\) changes sign.

Step2: Analyze the graph

• Concave - up interval:

By looking at the graph, we can see that the function is concave up on the interval \((-1,\infty)\). In this region, the graph “holds water” (a common mnemonic for concave up).

• Concave - down interval:

The function is concave down on the interval \((-\infty,-1)\). In this region, the graph “spills water” (a common mnemonic for concave down).

• Inflection point:

The inflection point occurs where the concavity changes. From the graph, when \(x = - 1\), the concavity changes from concave down to concave up. To find the \(y\) - value, we look at the graph. When \(x=-1\), \(y = 2\). So the inflection point is \((-1,2)\)

Answer:

Concave up on the interval(s): \((-1,\infty)\)
Concave down on the interval(s): \((-\infty,-1)\)
There is an inflection point at: \((-1,2)\)