Question

1. sketch ( f(x)=-x^{3}+4x^{2}-3x ).

Explanation:

Step1: Find the x - intercepts

Factor the function \(f(x)=-x^{3}+4x^{2}-3x=-x(x^{2}-4x + 3)=-x(x - 1)(x - 3)\).
Set \(y = 0\), then \(-x(x - 1)(x - 3)=0\).
Using the zero - product property \(a\times b\times c = 0\) implies \(a = 0\) or \(b = 0\) or \(c = 0\).
So \(x=0\), \(x = 1\), \(x = 3\) are the x - intercepts.

Step2: Find the y - intercept

Set \(x = 0\) in \(y=-x^{3}+4x^{2}-3x\).
Then \(y=f(0)=0\).

Step3: Analyze the end - behavior

Since the degree of the polynomial \(y=-x^{3}+4x^{2}-3x\) is \(n = 3\) (odd) and the leading coefficient \(a=-1\lt0\).
As \(x
ightarrow-\infty\), \(y=-x^{3}+4x^{2}-3x\approx -x^{3}
ightarrow+\infty\) (because \(-(-\infty)^{3}=+\infty\)).
As \(x
ightarrow+\infty\), \(y=-x^{3}+4x^{2}-3x\approx -x^{3}
ightarrow-\infty\).

Step4: Find the critical points

Differentiate \(y=-x^{3}+4x^{2}-3x\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(y^\prime=-3x^{2}+8x - 3\).
Set \(y^\prime = 0\), then \(3x^{2}-8x + 3=0\).
Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(ax^{2}+bx + c = 0\) (here \(a = 3\), \(b=-8\), \(c = 3\)).
\(x=\frac{8\pm\sqrt{64-36}}{6}=\frac{8\pm\sqrt{28}}{6}=\frac{8\pm2\sqrt{7}}{6}=\frac{4\pm\sqrt{7}}{3}\approx\frac{4\pm2.65}{3}\).
\(x_1=\frac{4+\sqrt{7}}{3}\approx2.22\), \(x_2=\frac{4-\sqrt{7}}{3}\approx0.78\).

Step5: Analyze the sign of the derivative

Take test intervals:
For \(x\lt\frac{4 - \sqrt{7}}{3}\approx0.78\), let \(x = 0\), then \(y^\prime(0)=-3(0)^{2}+8(0)-3=-3\lt0\).
For \(\frac{4 - \sqrt{7}}{3}\lt x\lt\frac{4+\sqrt{7}}{3}\approx2.22\), let \(x = 1\), then \(y^\prime(1)=-3(1)^{2}+8(1)-3=2\gt0\).
For \(x\gt\frac{4+\sqrt{7}}{3}\), let \(x = 3\), then \(y^\prime(3)=-3(3)^{2}+8(3)-3=-6\lt0\).

Answer:

Plot the x - intercepts \((0,0)\), \((1,0)\), \((3,0)\), y - intercept \((0,0)\). As \(x
ightarrow-\infty,y
ightarrow+\infty\) and as \(x
ightarrow+\infty,y
ightarrow-\infty\). The function is decreasing on \((-\infty,\frac{4 - \sqrt{7}}{3})\) and \((\frac{4+\sqrt{7}}{3},+\infty)\), increasing on \((\frac{4 - \sqrt{7}}{3},\frac{4+\sqrt{7}}{3})\). Sketch the curve passing through these points and with the determined behavior.