## The solution set of the inequality ({3^{3x}} – {5^{3x}} + 3left( {{3^x} – {5^x}} right) > 0) is – Math Book

The solution set of the inequality $${3^{3x}} – {5^{3x}} + 3\left( {{3^x} – {5^x}} \right) > 0$$ is
A. $$\left( { – \infty ;0} \right)$$. B. $$\left( { – \infty ;0} \right]$$. C. $$\left( {0; + \infty } \right)$$. D. $$\left[{0;+\infty}\right)$$[{0;+\infty}\right)\)
The given inequality is equivalent to $${3^{3x}} + {3.3^x} > {5^{3x}} + {3.5^x} \Leftrightarrow f\left( {{3^x}} \right) > f\left( {{5^x}} \right)$$.
Considering the function $$f\left( t \right) = {t^3} + 3t$$ on the interval $$\left( {0; + \infty } \right)$$, we have: $$f’ \left( t \right) = 3{t^2} + 3 > 0,\forall t > 0$$.
Derive a covariate function on the interval $$\left( {0; + \infty } \right)$$.
So from $$f\left( {{3^x}} \right) > f\left( {{5^x}} \right)$$ we have \({3^x} > {5^x } \Leftrightarrow {\left( {\frac{5}{3}} \right)^x}