Question

If \( R \to R \) be such that \( f(1) = 3 \) and \( f'(1) = 6 \), then \( f(x) \) is equal to

• A. \( e^x \)
• B. \( e^{x^2} \)
• C. \( e^{3x} \)
• D. \( e^{x^3} \)

Hint:

To find a function given its derivative and specific values, integrate the derivative and use initial conditions.

Answer 1

The Correct Option is C

Step 1: Analyzing the derivative.
Given that \( f'(x) = 6 \), the solution function that satisfies the derivative and the given values is \( e^{3x} \). The function \( f(x) = e^{3x} \) satisfies \( f(1) = 3 \) and \( f'(1) = 6 \).

Step 2: Conclusion.
The correct function is \( e^{3x} \), corresponding to option (3).