{"question":{"question":{"content":"Let $f(x)=x^6-3x^5$. **On which intervals is $f$ decreasing?** [[☃ radio 1]] ","images":{},"widgets":{"radio 1":{"type":"radio","alignment":"default","static":false,"graded":true,"options":{"choices":[{"content":"$x<0$ and $0\dfrac52$","correct":false},{"isNoneOfTheAbove":false,"content":"$x<0$ only","correct":false},{"isNoneOfTheAbove":false,"content":"All real numbers (excluding $0$ and $\dfrac{5}{2}$ )","correct":false}],"randomize":false,"multipleSelect":false,"displayCount":null,"hasNoneOfTheAbove":false,"onePerLine":true,"deselectEnabled":false},"version":{"major":1,"minor":0}}}},"answerArea":{"calculator":false,"chi2Table":false,"periodicTable":false,"tTable":false,"zTable":false},"itemDataVersion":{"major":0,"minor":1},"hints":[{"replace":false,"content":"We can analyze the intervals where $f$ is increasing/decreasing by looking for the intervals where its derivative $f'$ is positive/negative. A function can only change its direction (from increasing to decreasing and vice versa) between its **critical points**. So let's find the critical points of $f$. [[☃ explanation 1]]","images":{},"widgets":{"explanation 1":{"type":"explanation","alignment":"default","static":false,"graded":true,"options":{"static":false,"showPrompt":"Remind me what critical points are?","hidePrompt":"Got it, thanks!","explanation":"The critical points, or critical numbers, of a function $u$ are the $x$-values for which $u'(x)=0$ or where $u'$ isn't defined.","widgets":{}},"version":{"major":0,"minor":0}}}},{"replace":false,"content":"The derivative of $f$ is $f'(x)=3x^4(2x-5)$. [[☃ explanation 1]] - $f'(x)=0$ for $x=0,\dfrac52$. - Since $f'$ is a polynomial, it's defined for all real numbers. Therefore, our critical points are $x=0$ and $x=\dfrac52$.","images":{},"widgets":{"explanation 1":{"type":"explanation","alignment":"default","static":false,"graded":true,"options":{"static":false,"showPrompt":"Show me how you found that.","hidePrompt":"Got it, thanks!","explanation":"$\begin{align} f'(x)&=\dfrac{d}{dx}(x^6-3x^5) \\\\ &=\dfrac{d}{dx}(x^6)-3\dfrac{d}{dx}(x^5) \\\\ &=6x^5-3(5x^4) \\\\ &=6x^5-15x^4 \\\\ &=3x^4(2x-5) \end{align}$","widgets":{}},"version":{"major":0,"minor":0}}}},{"replace":false,"content":"Our critical points divide the number line into three intervals: ![number line](web+graphie://ka-perseus-graphie.s3.amazonaws.com/e0e97643317d732141b331cab3a537cbe502770f) Let's evaluate $f'$ at each interval to see if it's positive or negative on that interval. Interval | $x$-value | $f'(x)$ | Verdict :-: | :-: | :-: | :-: $x<0$ | $x=-1$ | $f'(-1)=-21<0$ | $f$ is decreasing $\searrow$ $0\dfrac{5}{2}$ | $x=3$ | $f'(3)=243>0$ | $f$ is increasing $\nearrow$","images":{"web+graphie://ka-perseus-graphie.s3.amazonaws.com/e0e97643317d732141b331cab3a537cbe502770f":{"width":460,"height":80}},"widgets":{}},{"replace":false,"content":"In conclusion, $f$ is decreasing over the intervals $x<0$ and $0

Increasing & decreasing intervals

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