3 for each of the following functions:\n i find and classify all turning points.\n ii find and classify all…

For function $f(x)=x^{2}-5x + 4$

Explanation:

Step1: Find the first - derivative

The derivative of $f(x)=x^{2}-5x + 4$ using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $f^\prime(x)=2x-5$.

Step2: Find turning points

Set $f^\prime(x)=0$, so $2x - 5=0$. Solving for $x$ gives $x=\frac{5}{2}$. Substitute $x = \frac{5}{2}$ into $f(x)$: $f(\frac{5}{2})=(\frac{5}{2})^{2}-5\times\frac{5}{2}+4=\frac{25}{4}-\frac{25}{2}+4=\frac{25 - 50 + 16}{4}=-\frac{9}{4}$. The second - derivative $f^{\prime\prime}(x) = 2>0$, so the turning point $(\frac{5}{2},-\frac{9}{4})$ is a minimum.

Step3: Find the second - derivative

$f^{\prime\prime}(x)=2$. Since $f^{\prime\prime}(x)$ is a non - zero constant, there are no points of inflection.

Step4: Find intervals of increase and decrease

$f^\prime(x)=2x - 5$. When $f^\prime(x)>0$, $2x-5>0$, $x>\frac{5}{2}$, so the function is increasing on the interval $(\frac{5}{2},\infty)$. When $f^\prime(x)<0$, $2x - 5<0$, $x<\frac{5}{2}$, so the function is decreasing on the interval $(-\infty,\frac{5}{2})$.

Step5: Find intervals of concavity

Since $f^{\prime\prime}(x)=2>0$, the function is concave up on $(-\infty,\infty)$.

Step6: Sketch the function

The function is a parabola opening upwards (because $a = 1>0$ in $y=ax^{2}+bx + c$) with a minimum at $(\frac{5}{2},-\frac{9}{4})$, $y$ - intercept at $f(0)=4$, and $x$ - intercepts by setting $y = 0$: $x^{2}-5x + 4=(x - 1)(x - 4)=0$, so $x = 1$ and $x = 4$.

Answer:

• Turning point: $(\frac{5}{2},-\frac{9}{4})$, minimum.
• No points of inflection.
• Increasing on $(\frac{5}{2},\infty)$, decreasing on $(-\infty,\frac{5}{2})$.
• Concave up on $(-\infty,\infty)$.
• Sketch: Parabola opening upwards with $x$ - intercepts at $x = 1$ and $x = 4$, $y$ - intercept at $y = 4$ and minimum at $(\frac{5}{2},-\frac{9}{4})$.

We can follow a similar process for the other functions in the problem set. For example, for $f(x)=x^{3}+4x^{2}$:

Explanation:

Step1: Find the first - derivative

Using the power rule, $f^\prime(x)=3x^{2}+8x=x(3x + 8)$.

Step2: Find turning points

Set $f^\prime(x)=0$, then $x(3x + 8)=0$. So $x = 0$ or $x=-\frac{8}{3}$. $f(0)=0$ and $f(-\frac{8}{3})=(-\frac{8}{3})^{3}+4\times(-\frac{8}{3})^{2}=-\frac{512}{27}+\frac{256}{9}=\frac{-512 + 768}{27}=\frac{256}{27}$. The second - derivative $f^{\prime\prime}(x)=6x + 8$. $f^{\prime\prime}(0)=8>0$, so $(0,0)$ is a minimum. $f^{\prime\prime}(-\frac{8}{3})=6\times(-\frac{8}{3})+8=-16 + 8=-8<0$, so $(-\frac{8}{3},\frac{256}{27})$ is a maximum.

Step3: Find the second - derivative

$f^{\prime\prime}(x)=6x + 8$. Set $f^{\prime\prime}(x)=0$, then $6x+8 = 0$, $x=-\frac{4}{3}$. $f(-\frac{4}{3})=(-\frac{4}{3})^{3}+4\times(-\frac{4}{3})^{2}=-\frac{64}{27}+\frac{64}{9}=\frac{-64 + 192}{27}=\frac{128}{27}$. So the point of inflection is $(-\frac{4}{3},\frac{128}{27})$.

Step4: Find intervals of increase and decrease

$f^\prime(x)=x(3x + 8)$. When $f^\prime(x)>0$, the solution of the inequality $x(3x + 8)>0$ is $x<-\frac{8}{3}$ or $x>0$. When $f^\prime(x)<0$, the solution of the inequality $x(3x + 8)<0$ is $-\frac{8}{3}<x<0$.

Step5: Find intervals of concavity

When $f^{\prime\prime}(x)=6x + 8>0$, $x>-\frac{4}{3}$, the function is concave up. When $f^{\prime\prime}(x)=6x + 8<0$, $x<-\frac{4}{3}$, the function is concave down.

Step6: Sketch the function

Mark the turning points $(-\frac{8}{3},\frac{256}{27})$ (maximum), $(0,0)$ (minimum), the point of inflection $(-\frac{4}{3},\frac{128}{27})$, and use the information about intervals of increase/decrease and concavity to draw the curve.

Answer:

• Turning points: $(-\frac{8}{3},\frac{256}{27})$ (maximum), $(0,0)$ (minimum).
• Point of inflection: $(-\frac{4}{3},\frac{128}{27})$.
• Increasing on $(-\infty,-\frac{8}{3})\cup(0,\infty)$, decreasing on $(-\frac{8}{3},0)$.
• Concave up on $(-\frac{4}{3},\infty)$, concave down on $(-\infty,-\frac{4}{3})$.
• Sketch: Mark the key points and draw the curve according to the behavior.

The remaining functions can be analyzed in the same step - by - step manner using derivatives to find turning points, points of inflection, intervals of increase/decrease and concavity.