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 first - derivative

The function is \(y =-\frac{x^{3}}{3}+x^{2}\). Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the first - derivative \(y^\prime=-x^{2}+2x=-x(x - 2)\).

Step2: Find the critical points

Set \(y^\prime = 0\). Then \(-x(x - 2)=0\). Solving for \(x\), we get \(x = 0\) and \(x = 2\).

Step3: Determine the intervals of increase and decrease

We use test intervals. The intervals are \((-\infty,0)\), \((0,2)\) and \((2,\infty)\).

• For \(x=-1\) (in the interval \((-\infty,0)\)), \(y^\prime=-(-1)(-1 - 2)=-3<0\). So the function is decreasing on \((-\infty,0)\).
• For \(x = 1\) (in the interval \((0,2)\)), \(y^\prime=-1(1 - 2)=1>0\). So the function is increasing on \((0,2)\).
• For \(x = 3\) (in the interval \((2,\infty)\)), \(y^\prime=-3(3 - 2)=-3<0\). So the function is decreasing on \((2,\infty)\).

Step4: Find the relative minima and maxima

Using the first - derivative test:

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

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 \(-2(x - 1)=0\), so \(x = 1\). When \(x = 1\), \(y(1)=-\frac{1^{3}}{3}+1^{2}=\frac{2}{3}\). The inflection point is \((1,\frac{2}{3})\).

Step7: Find the \(y\) - intercept

Set \(x = 0\), \(y=0\). So the \(y\) - intercept is \((0,0)\).

Step8: Sketch the graph

• Plot the \(y\) - intercept \((0,0)\), the relative minimum \((0,0)\), the relative maximum \((2,\frac{4}{3})\) and the inflection point \((1,\frac{2}{3})\).
• Use the intervals of increase/decrease: decreasing on \((-\infty,0)\), increasing on \((0,2)\) and decreasing on \((2,\infty)\) to sketch the curve.

Answer:

• \(y\) - intercept: \((0,0)\)
• Critical points: \(x = 0\) (relative minimum) and \(x = 2\) (relative maximum)
• Intervals of increase: \((0,2)\)
• Intervals of decrease: \((-\infty,0)\cup(2,\infty)\)
• Inflection point: \((1,\frac{2}{3})\)