Question

for each problem, find the x and y intercepts, x - coordinates of the critical points, open intervals where the function is increasing and decreasing, x - coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. using this information, sketch the graph of the function.
( y = \frac{x^4}{4} + x^3 )

Explanation:

Step1: Find \(y -\)intercept

Set \(x = 0\) in \(y=-\frac{x^{3}}{3}+x^{2}\).
\(y=-\frac{0^{3}}{3}+0^{2}=0\). So, \(y -\)intercept is \((0,0)\).

Step2: Find critical points

Differentiate \(y\) with respect to \(x\). \(y^{\prime}=-x^{2}+2x\).
Set \(y^{\prime}=0\), then \(-x^{2}+2x = 0\). Factor out \(-x\): \(-x(x - 2)=0\).
Solutions are \(x = 0\) and \(x = 2\).

Step3: Determine intervals of increase and decrease

Use test - points.
For the interval \((-\infty,0)\), let \(x=-1\). Then \(y^{\prime}=-(-1)^{2}+2(-1)=-1 - 2=-3<0\). So, \(y\) is decreasing on \((-\infty,0)\).
For the interval \((0,2)\), let \(x = 1\). Then \(y^{\prime}=-1^{2}+2\times1=1>0\). So, \(y\) is increasing on \((0,2)\).
For the interval \((2,\infty)\), let \(x = 3\). Then \(y^{\prime}=-3^{2}+2\times3=-9 + 6=-3<0\). So, \(y\) is decreasing on \((2,\infty)\).
Relative minimum at \(x = 0\): \(y=-\frac{0^{3}}{3}+0^{2}=0\). Relative maximum at \(x = 2\): \(y=-\frac{2^{3}}{3}+2^{2}=-\frac{8}{3}+4=\frac{4}{3}\).

Step4: Find inflection points

Differentiate \(y^{\prime}=-x^{2}+2x\) to get \(y^{\prime\prime}=-2x + 2\).
Set \(y^{\prime\prime}=0\), then \(-2x+2 = 0\), \(x = 1\).
When \(x = 1\), \(y=-\frac{1^{3}}{3}+1^{2}=\frac{2}{3}\). So, inflection point is \((1,\frac{2}{3})\).
For \(x<1\), let \(x = 0\), \(y^{\prime\prime}=-2\times0 + 2=2>0\), function is concave up on \((-\infty,1)\).
For \(x>1\), let \(x = 2\), \(y^{\prime\prime}=-2\times2+2=-2<0\), function is concave down on \((1,\infty)\).

Answer:

• \(y -\)intercept: \((0,0)\)
• Critical points: \(x = 0\) (relative minimum, \(y = 0\)) and \(x = 2\) (relative maximum, \(y=\frac{4}{3}\))
• Increasing interval: \((0,2)\)
• Decreasing intervals: \((-\infty,0)\cup(2,\infty)\)
• Inflection point: \((1,\frac{2}{3})\)
• Concave up interval: \((-\infty,1)\)
• Concave down interval: \((1,\infty)\)