{"question":{"question":{"content":"**Select all the intervals where $g'(x)<0$ and $g''(x)>0$.** [[☃ radio 1]] ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/131fcf44bfaaf28ae62f06712b9ca462b6b29742) ","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/131fcf44bfaaf28ae62f06712b9ca462b6b29742":{"width":425,"height":425}},"widgets":{"radio 1":{"type":"radio","alignment":"default","static":false,"graded":true,"options":{"choices":[{"content":"$-40$ means $g$ is **increasing**. $g'(x)<0$ means$g$ is **decreasing**. $g''(x)>0$ means the slope of $g$ is increasing ($g$ is **concave up**). $g''(x)<0$ means the slope of $g$ is decreasing ($g$ is **concave down**). | | $g'(x)>0$ | $g'(x)<0$ :- | :- | :-: | :-: | | $g$ is **increasing** | $g$ is **decreasing** $g''(x)>0$ | $g$ is **concave up** | ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/fad1dc95cd761b6e7f4c581c7093841fcf16029a) | ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/88e8d1ff71cbe909ea256814c06e1ff2df39bd5e) $g''(x)<0$ | $g$ is **concave down** | ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/f9425dbee4d4f73f6188a62f88f637018843d7c8) | ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/4d64d76d6423a90abf605dfe2b856e7327f92a0c) We are asked to find an interval where $g'(x)<0$ and $g''(x)>0$, so we need an interval where the graph of $g$ has this general shape: ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/88e8d1ff71cbe909ea256814c06e1ff2df39bd5e)","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/fad1dc95cd761b6e7f4c581c7093841fcf16029a":{"width":100,"height":100},"web+graphie://ka-perseus-graphie.s3.amazonaws.com/f9425dbee4d4f73f6188a62f88f637018843d7c8":{"width":100,"height":100},"web+graphie://ka-perseus-graphie.s3.amazonaws.com/88e8d1ff71cbe909ea256814c06e1ff2df39bd5e":{"width":100,"height":100},"web+graphie://ka-perseus-graphie.s3.amazonaws.com/4d64d76d6423a90abf605dfe2b856e7327f92a0c":{"width":100,"height":100}},"widgets":{}},{"replace":false,"content":"Visually, this is where $g'(x)<0$ and $g''(x)>0$. ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/6cbcda5ba8cbd0429787e9dbc0a5d41ef8e49cb8)","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/6cbcda5ba8cbd0429787e9dbc0a5d41ef8e49cb8":{"width":425,"height":425}},"widgets":{}},{"replace":false,"content":"Now we can check each option: Interval | Shape | Is $g'(x)<0$ and $g''(x)>0$? :-: | :-: | :-: $-40$ but $g'(x)>0$. $3

Analyze concavity (graphical)

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