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 \(y -\)intercept

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

Step2: Find the first - derivative

Differentiate \(y =-\frac{x^{3}}{3}+x^{2}\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(y^\prime=-x^{2}+2x=-x(x - 2)\)

Step3: Find the critical points

Set \(y^\prime = 0\), so \(-x(x - 2)=0\).
Solving \(x(x - 2)=0\) gives \(x = 0\) and \(x = 2\)

Step4: Determine the intervals of increase and decrease

Use a sign - chart for \(y^\prime\).
Choose test points:

• For \(x\lt0\), let \(x=-1\). Then \(y^\prime=-(-1)(-1 - 2)=-3\lt0\), so the function is decreasing on \((-\infty,0)\)
• For \(0\lt x\lt2\), let \(x = 1\). Then \(y^\prime=-1(1 - 2)=1\gt0\), so the function is increasing on \((0,2)\)
• For \(x\gt2\), let \(x = 3\). Then \(y^\prime=-3(3 - 2)=-3\lt0\), so the function is decreasing on \((2,\infty)\)

Step5: Find the second - derivative

Differentiate \(y^\prime=-x^{2}+2x\). Using the power rule, \(y^{\prime\prime}=-2x + 2=-2(x - 1)\)

Step6: Find the inflection point

Set \(y^{\prime\prime}=0\), then \(-2(x - 1)=0\), which gives \(x = 1\).
When \(x = 1\), \(y=-\frac{1^{3}}{3}+1^{2}=\frac{2}{3}\). So the inflection point is \((1,\frac{2}{3})\)

Step7: Find the relative minima and maxima

Since the function changes from decreasing \((x\lt0)\) to increasing \((0\lt x\lt2)\) at \(x = 0\), \(y(0)=0\) is a relative minimum.
Since the function changes from increasing \((0\lt x\lt2)\) to decreasing \((x\gt2)\) at \(x = 2\), \(y(2)=-\frac{2^{3}}{3}+2^{2}=-\frac{8}{3}+4=\frac{4}{3}\) is a relative maximum

Answer:

• \(y -\)intercept: \((0,0)\)
• Critical points: \(x = 0\) and \(x = 2\)
• Increasing on \((0,2)\), decreasing on \((-\infty,0)\cup(2,\infty)\)
• Inflection point: \((1,\frac{2}{3})\)
• Relative minimum at \((0,0)\), relative maximum at \((2,\frac{4}{3})\)

To sketch the graph:

• Plot the \(y -\)intercept \((0,0)\), inflection point \((1,\frac{2}{3})\), relative minimum \((0,0)\) and relative maximum \((2,\frac{4}{3})\)
• Use the intervals of increase and decrease to draw the curve. The function is concave down when \(y^{\prime\prime}\lt0\) (i.e., \(x\gt1\)) and concave up when \(y^{\prime\prime}\gt0\) (i.e., \(x\lt1\))