Question

PLEASE PLEASE PLEASE PLEASE PLEASE HELP ME PLEASE HELP!?!?!!?!?!?!?!!?!!?!

The balance in two separate bank accounts grows each month at different rates. The growth rates for both accounts are represented by the functions f(x) = 2^x and g(x) = 4x + 12. In what month is the f(x) balance greater than the g(x) balance?

Select one:
a. Month 6
b. Month 5
c. Month 4
d. Month 3

Answer 1

`\text{Goal: Find } x \text{ such that:} \\ 2^x > 4x + 12`


`2^x > 4x + 12 \\ 2^x > 4(x + 3) \\ 2^(x - 2) > x + 3`


`\text{A lemma: } \\ \text{It is given that: } \\ 2^n > 2n + 1, \text{ } n > 3`

`\text{This can be proven by Mathematical Induction.}`

`\text{Using this lemma, we can manipulate the expression:} \\ 2^(x - 2) > 2(x - 2) + 1 = 2x - 3, \text{ } x > 5 \\ 2\cdot 2^(x - 2) > 2(2x - 3) \\ 2^(x - 1) > 4x - 12 \\ 2^(x - 1) + 24 > 4x + 12`


`\text{Observe the following: } 2^x \text{ will always grow faster than } 2^(x - 1) \\ \text{This implies that } 2^(x) > 2^(x - 1) + 24 > 4x + 12, \text{ given that } x > 5`


`\text{This means that the first month } f(x) > g(x) \text{ is Month 6.}`