{"question":{"question":{"content":"**Select all the intervals where $h'(x)>0$ and $h''(x)<0$.** [[☃ radio 1]] ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/478c88849a71a04bdbc4b23c7a245dfdd39c35fd) ","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/478c88849a71a04bdbc4b23c7a245dfdd39c35fd":{"width":425,"height":425}},"widgets":{"radio 1":{"type":"radio","alignment":"default","static":false,"graded":true,"options":{"choices":[{"content":"$-4.50$ means $h$ is **increasing**. $h'(x)<0$ means$h$ is **decreasing**. $h''(x)>0$ means the slope of $h$ is increasing ($h$ is **concave up**). $h''(x)<0$ means the slope of $h$ is decreasing ($h$ is **concave down**). | | $h'(x)>0$ | $h'(x)<0$ :- | :- | :-: | :-: | | $h$ is **increasing** | $h$ is **decreasing** $h''(x)>0$ | $h$ is **concave up** | ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/fad1dc95cd761b6e7f4c581c7093841fcf16029a) | ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/88e8d1ff71cbe909ea256814c06e1ff2df39bd5e) $h''(x)<0$ | $h$ 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 $h'(x)>0$ and $h''(x)<0$, so we need an interval where the graph of $h$ has this general shape: ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/f9425dbee4d4f73f6188a62f88f637018843d7c8)","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 $h'(x)>0$ and $h''(x)<0$. ![](web+graphie://ka-perseus-graphie.s3.amazonaws.com/5de26007755a4be198a06911ad3a8d0a961d31a9)","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/5de26007755a4be198a06911ad3a8d0a961d31a9":{"width":425,"height":425}},"widgets":{}},{"replace":false,"content":"Now we can check each option: Interval | Shape | Is $h'(x)>0$ and $h''(x)<0$? :-: | :-: | :-: $-4.50$. $-10$ but $h''(x)>0$.","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/08e8b7e5ce7d160e69c506cebdd2552cb4f57715":{"width":50,"height":50},"web+graphie://ka-perseus-graphie.s3.amazonaws.com/1423ac842506af6991b8e635c34a9d36faf33368":{"width":50,"height":50},"web+graphie://ka-perseus-graphie.s3.amazonaws.com/842593fcfd6a52a51cd66950f88ff7cd344d7af3":{"width":50,"height":50}},"widgets":{}}]},"problemId":39529,"exerciseId":3264,"exerciseSessionId":null,"hintsEnabled":true,"readOnly":false,"submitterId":null,"checkAnswerButtonText":"Check answer","correctButtonText":"🌟 Yes! You got it!","tryAgainButtonText":"Try again"}

Analyze concavity (graphical)