Question

Show that the function given by \( f(x) = e^{3x} \) is increasing on \( R \).

Hint:

For exponential functions with positive exponents, the function is always increasing as long as the exponent's coefficient is positive.

Answer 1

Step 1: Understand the function.
The function is \( f(x) = e^{3x} \). For a function to be increasing, its derivative must be positive.

Step 2: Differentiate the function.
The derivative of \( f(x) = e^{3x} \) with respect to \( x \) is: \[ f'(x) = \frac{d}{dx} (e^{3x}) = 3e^{3x}. \]

Step 3: Analyze the derivative.
Since \( e^{3x} \) is always positive for all values of \( x \), and the constant factor \( 3 \) is also positive, we have: \[ f'(x) = 3e^{3x} > 0 \text{for all} x \in R. \]

Step 4: Conclusion.
Thus, the function \( f(x) = e^{3x} \) is increasing on \( R \).