Answer
See the explanation
Work Step by Step
Certainly! If you know the derivative \( f'(x) \) and the value of \( f \) at a specific point \( x_0 \), you can use this information to identify the original function \( f(x) \) by finding the antiderivative (indefinite integral) of \( f'(x) \) with respect to \( x \). The antiderivative of \( f'(x) \) will give you \( f(x) \) up to a constant.
Example:
If \( f'(x) = 2x \) and you know that \( f(1) = 3 \), you integrate \( f'(x) \) with respect to \( x \) to find \( f(x) \):
\[ \int 2x \,dx = x^2 + C \]Now, use the given point \( f(1) = 3 \) to solve for the constant \( C \):\[ (1)^2 + C = 3 \]\[ C = 2 \]
Therefore, \( f(x) = x^2 + 2 \).
