Nature of stationary point

$$\begin{array}{rl}{f}^{″}(x)& >0\text{local minimum}\\ \\ f^{″}(x)& <0\text{local maximum}\end{array}$$

Decreasing function

$f^{\prime }(x)<0$

Curve sketching

• Determine curve intercept with axis
• Determine stationary points (if exists)
• Determine nature of stationary points (if exists)

Example

Given a function $f(x)=x^4-2x^3+x^2$

A. Determine its stationary points and its nature

B. Find the inverval where the function is increasing and decreasing

C. Sketch the curve

A. Stationary points

$$\begin{array}{rl}& f^{\prime }(x)=0\\ \\ & 4x^3-6x^2+2x=0\\ \\ & 2x.(2x^2-3x+1)=0\\ \\ & 2x.(2x-1)(x-1)=0\\ \\ & x=0\text{ or }x=\frac{1}{2}\text{ or }x=1\end{array}$$

Nature of stationary points

$$\begin{array}{rl}& f^{″}(x)=12x^2-12x+2\\ \\ & f^{″}(0)=2\text{(local minimum)}\\ \\ & f^{″}(\frac{1}{2})=-1\text{(local maximum)}\\ \\ & f^{″}(1)=2\text{(local minimum)}\end{array}$$

Coordinate of stationary points

$$\begin{array}{rl}& f(x)=x^4-2x^3+x^2\\ \\ & f(0)=0\\ \\ & f\left(\frac{1}{2}\right)=\frac{1}{16}\\ \\ & f(1)=0\end{array}$$

Coordinate of stationary points are:

$(0,0)$ as local minimum

$({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{16}})$ as local maximum

$(1,0)$ as local minimum

B. Increasing and decreasing interval

Increasing function at the interval $0<x<\frac{1}{2}$ dan $x>1$

Decreasing function at the interval $x<0$ dan $\frac{1}{2}<x<1$

C. Curve sketching