Contoh Soal 4 Turunan Fungsi Aljabar

 $\begin{array}{l}\\ 16.& \text{Jika grafik fungsi}f(x)=5+15x+9x^2+x^3\\ & \text{naik untuk}x\text{yang memenuhi}....\\ & \begin{array}{l}\\ \text{a}.x<1\text{atau}x>5& \text{d}.x<-5\text{atau}x>-1\\ \text{b}.-{\displaystyle 1<x<5}& \text{e}.-5<x<1\\ \text{c}.{\displaystyle -5<x<-1}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& \text{Diketahui bahwa}:\\ & f(x)=x^3+9x^2+15x+5.\\ & \text{Dikatakan fungsi}f\text{naik, maka}\\ & f{}^{\prime }(x)>0\\ & \iff 3x^2+18x+15>0,\text{tiap ruas dibagi 3}\\ & \iff x^2+6x+5>0\\ & \iff (x+1)(x+5)>0\end{array}\\ & \text{Berikut ilustrasi gambarnya}\end{array}$.

$\begin{array}{l}\\ 17.& \text{Sebuah bola dilempar ke atas secara vertikal.}\\ & \text{Jika lintasan bola pada saat}t\text{detik adalah}\\ & h(t)=-{\displaystyle \frac{1}{4}{t}^4+\frac{2}{3}{t}^3+4t^2+5\text{m},}\\ & \text{maka tinggi maksimum yang}\\ & \text{dicapai oleh bola tersebut adalah}....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{67}{3}}& & \text{d}.{\displaystyle \frac{133}{3}}\\ \\ \text{b}.{\displaystyle \frac{123}{3}}& \text{c}.{\displaystyle \frac{128}{3}}& \text{e}.{\displaystyle \frac{143}{3}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{Perh}& \text{atikan bahwa lintasan bola saat dilempar }\\ \text{verti}& \text{kal dituliskan dengan fungsi}\\ h(t)& =-{\displaystyle \frac{1}{4}{t}^4+\frac{2}{3}{t}^3+4t^2+5.}\\ \text{mak}& \text{a tinggi maksimum akan dicapai bola }\\ \text{saat}& h{}^{\prime }(t)=0,\\ \text{Sela}& \text{njutnya},\\ h{}^{\prime }(t)& =0\\ -t^3+& 2t^2+8t=0\\ -t(t^2& -2t-8)=0\\ t(t-& 4)(t+2)=0\{\begin{array}{ll}t& =0\\ t& =4\\ t& =-2\end{array}\\ \text{kita}& \text{ambil yang bernilai positif untuk }\\ & t\text{yaitu}t=4.\\ t=4& \to h(4)=-{\displaystyle \frac{1}{4}(4{)}^4+\frac{2}{3}(4{)}^3+4(4{)}^2+5}\\ h(4)& =-64+{\displaystyle \frac{128}{3}+64+5}\\ & ={\displaystyle \frac{143}{3}m}\end{array}\\ & \text{Berikut ilustrasi gambarnya}\end{array}$.