Titik Stationer

Titik Stationer

TITIK STATIONER

Titik stationer

\begin{equation*} f'(x) = 0 \end{equation*}

Jenis titik stationer

\begin{equation*} \begin{split} f''(x) & > 0 \quad \text{titik minimum lokal} \\\\ f''(x) & < 0 \quad \text{titik maksimum lokal} \\\\ f''(x) & = 0 \quad \text{titik belok} \end{split} \end{equation*}

Contoh

Diketahui kurva $$f(x)=\dfrac{1}{5} x^{5}-\dfrac{1}{3} x^{3}$$.

Tentukan titik-titik stasioner kurva dan jenisnya.

Koordinat titik stasioner

\begin{equation*}
\begin{split}
& f(x)=\frac{1}{5} x^{5}-\frac{1}{3} x^{3} \\\\
& f' (x) =0\\\\
& x^{4}-x^{2} =0\\\\
& x^{2} \:.\: (x^{2}-1) =0\\\\
& x^2 \:.\: (x+1) \:.\: (x-1) =0\\\\
& x=0 \text{ atau } x=-1 \text{ atau } x=1
\end{split}
\end{equation*}

Titik x = −1

\begin{equation*} \begin{split} f(-1) & = \frac{1}{5} (-1)^{5}-\frac{1}{3} (-1)^{3} \\\\ f(-1) & = \frac{2}{15} \end{split} \end{equation*}

Koordinat titik $$\left(-1,\dfrac{2}{15} \right)$$

Titik x = 0

\begin{equation*} \begin{split} f(0) & = \frac{1}{5} (0)^{5}-\frac{1}{3} (0)^{3} \\\\ f(0) & = 0 \\\\ \end{split} \end{equation*}

Koordinat titik $$(0,0)$$

Titik x = 1

\begin{equation*} \begin{split} f(1) & = \frac{1}{5} (1)^{5}-\frac{1}{3} (1)^{3} \\\\ f(1) & = -\frac{2}{15} \end{split} \end{equation*}

Koordinat titik $$\left(1,-\dfrac{2}{15} \right)$$

Menentukan jenis titik stasioner dengan uji turunan kedua:

\begin{equation*}
\begin{split}
f(x) & =\frac{1}{5} x^{5}-\frac{1}{3} x^{3} \\\\
f'(x) & = x^{4}-x^{2} \\\\
f''(x)&=4x^{3}-2x
\end{split}
\end{equation*}

Titik $$\left (-1,\dfrac{2}{15} \right)$$

\begin{equation*} \begin{split} f''(-1) & = 4(-1)^{3}-2(-1) \\\\ f''(-1) & =-2 \: {\color[RGB]{0,0,255} < 0 \quad \text{(maksimum)}} \end{split} \end{equation*}

Titik $$(0,0)$$

\begin{equation*} \begin{split} f''(0)&=4(0)^{3}-2(0) \\\\ f''(0)& =0 \quad {\color[RGB]{0,0,255} \text{(titik belok)}} \end{split} \end{equation*}

Titik $$\left(1,-\dfrac{2}{15} \right)$$

\begin{equation*} \begin{split} f''(1) & =4(1)^{3}-2(1) \\\\ f''(1) & = 2 \: {\color[RGB]{0,0,255} > 0 \quad \text{(minimum)}} \end{split} \end{equation*}