The domain of the funciton `f(x)` given by `3^(x) + 3^(f) = "min" (2t^(3) - 15t^(2) + 36+ - 25, 2 + |sin t| , 2 le t le 4)` is

Options

A. `(-oo, 1 )`
B. `(-oo, log_3 e )`
C. `(0, log_3 2 )`
D. `(-oo, log_3 2 )`

Answer

D

Explanation

Let `g(t) = 2t^(3) - 15t^(2) + 36t - 25`. Then,
`g'(t) = 6t^(2) - 30t + 3 rArr g' (t) = 6(t-2) (t-3)`
`therefore g' (t) = 0 rArr t= 2,3`.
Signs of g'(t) for different values of t are shown in Fig. 18
Clearly, g attains local minimun value at t = 3.

`therefore { g(t)_("min") = g(3) = 2`
Also, `2 le 2 + |sin t| le 3`
Hence, min `{2t^(3) - 15t^(2) + 36t - 25, 2+ |sin t|, 2 le t le 4}=2`
`therefore 3^(x) + 3^(f(x)) = 2 - 3^(x) rArr f(x) = log_(3)(2-3^(x))`
For f(x) to be real, we must have
`2-3^(x) gt = 3^(x) lt 2 rArr x log_(3) 2 rArr x in(-oo, log_(3) 2)`