Answer:
From fourth month onwards, the growth rate of [tex]f(x)[/tex] is greater than that of [tex]g(x)[/tex].
Step-by-step explanation:
Given:
The growth rates of both bank accounts are given as:
[tex]f(x)=3^x\\g(x)=5x+25[/tex]
Now, as per question, we need to find the value of 'x' when the value of [tex]f(x)>g(x)[/tex]. Or,
[tex]3^x>5x+25[/tex]
Now, we can do this by checking the values of 'x' by hit and trial method.
Let [tex]x=1[/tex]. The inequality becomes:
[tex]3^1>5(1)+25\\3>30(False)[/tex]
Let [tex]x=2[/tex]. The inequality becomes:
[tex]3^2>5(2)+25\\9>35(False)[/tex]
Let [tex]x=3[/tex]. The inequality becomes:
[tex]3^3>5(3)+25\\27>40(False)[/tex]
Let [tex]x=4[/tex]. The inequality becomes:
[tex]3^4>5(4)+25\\81>45(True)[/tex]
Therefore, the value of 'x' for which [tex]f(x)>g(x)[/tex] is 4.
So, from the fourth month onwards, the balance in [tex]f(x)[/tex] becomes greater than [tex]g(x)[/tex].
The graphical solution is shown below to support the same.
From the graph, we can conclude that after the 'x' value equals 3.4, the graph of [tex]f(x)[/tex] lies above of [tex]g(x)[/tex]. Hence, [tex]f(x)>g(x)[/tex] for [tex]x>3.4[/tex]
