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^{2}

Explanation:

Step1: Find the \(x -\)intercepts

Set \(y = 0\), so \(\frac{-x^{3}}{3}+x^{2}=0\). Factor out \(-\frac{x^{2}}{3}\): \(-\frac{x^{2}}{3}(x - 3)=0\).
Using the zero - product property \(x = 0\) 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 \(y^\prime = 0\): \(-x^{2}+2x=0\). Factor out \(-x\): \(-x(x - 2)=0\). So \(x = 0\) or \(x = 2\).

• For \(x\lt0\), let \(x=-1\), then \(y^\prime=-(-1)^{2}+2(-1)=-1 - 2=-3\lt0\).
• For \(0\lt x\lt2\), let \(x = 1\), then \(y^\prime=-1^{2}+2\times1=1\gt0\).
• For \(x\gt2\), let \(x = 3\), then \(y^\prime=-3^{2}+2\times3=-9 + 6=-3\lt0\).

The function is decreasing on \((-\infty,0)\cup(2,\infty)\) and increasing on \((0,2)\).
At \(x = 0\), \(y=\frac{-0^{3}}{3}+0^{2}=0\); at \(x = 2\), \(y=\frac{-2^{3}}{3}+2^{2}=\frac{-8}{3}+4=\frac{4}{3}\). So the relative minimum is at \((0,0)\) and the relative maximum is at \((2,\frac{4}{3})\).

Step3: Find the second derivative

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

Answer:

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