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 the \(x -\)intercepts

Set \(y = 0\), so \(\frac{-x^{3}}{3}+x^{2}=0\). Factor out \(-\frac{x^{2}}{3}\), we get \(-\frac{x^{2}}{3}(x - 3)=0\).
Solving \(-\frac{x^{2}}{3}(x - 3)=0\) gives \(x = 0\) or \(x=3\).

Step2: Find the first - derivative

Differentiate \(y=\frac{-x^{3}}{3}+x^{2}\) using the power rule \(y^\prime=-x^{2}+2x\).
Set \(y^\prime = 0\), then \(-x^{2}+2x=0\). Factor out \(-x\), we have \(-x(x - 2)=0\).
Solving \(-x(x - 2)=0\) gives \(x = 0\) or \(x = 2\).
Create a sign chart for \(y^\prime\):

• Choose test points: for \(x\lt0\) (say \(x=-1\)), \(y^\prime=-(-1)^{2}+2(-1)=-3\lt0\).
• For \(0\lt x\lt2\) (say \(x = 1\)), \(y^\prime=-1^{2}+2\times1 = 1\gt0\).
• For \(x\gt2\) (say \(x=3\)), \(y^\prime=-3^{2}+2\times3=-3\lt0\).

The function is decreasing on \((-\infty,0)\cup(2,\infty)\) and increasing on \((0,2)\).
The critical points: when \(x = 0\), \(y=\frac{-0^{3}}{3}+0^{2}=0\); when \(x = 2\), \(y=\frac{-2^{3}}{3}+2^{2}=\frac{-8 + 12}{3}=\frac{4}{3}\). So the critical points are \((0,0)\) (a local minimum) and \((2,\frac{4}{3})\) (a local maximum).

Step3: Find the second - derivative

Differentiate \(y^\prime=-x^{2}+2x\) to get \(y^{\prime\prime}=-2x + 2\).
Set \(y^{\prime\prime}=0\), then \(-2x + 2=0\), which gives \(x = 1\).
When \(x\lt1\) (say \(x = 0\)), \(y^{\prime\prime}=-2\times0+2=2\gt0\), the function is concave up.
When \(x\gt1\) (say \(x = 2\)), \(y^{\prime\prime}=-2\times2+2=-2\lt0\), the function is concave down.
The inflection point: when \(x = 1\), \(y=\frac{-1^{3}}{3}+1^{2}=\frac{2}{3}\). So the inflection point is \((1,\frac{2}{3})\).

Answer:

• \(x -\)intercepts: \(x = 0\) and \(x = 3\).
• Critical points: \((0,0)\) (local minimum) and \((2,\frac{4}{3})\) (local maximum).
• Function is decreasing on \((-\infty,0)\cup(2,\infty)\) and increasing on \((0,2)\).
• Inflection point: \((1,\frac{2}{3})\).
• Concave up on \((-\infty,1)\) and concave down on \((1,\infty)\).