1. If an equation of the tangent line to the curve \( y=f(x) \) at the point where \( a=2 \) is \( y=4 x-5 \), find \( f(2) \) and \( f^{\prime}(2) \).

To determine \( f(2) \) and \( f^{\prime}(2) \) from the equation of the tangent line \( y = 4x - 5 \) to the curve \( y = f(x) \) at \( x = 2 \), we can follow these steps: 1. **Equation of the Tangent Line:** The general form of the tangent line to \( y = f(x) \) at \( x = a \) is: \[ y = f(a) + f^{\prime}(a)(x - a) \] Given \( a = 2 \), the tangent line equation becomes: \[ y = f(2) + f^{\prime}(2)(x - 2) \] 2. **Compare with the Given Tangent Line:** The given tangent line equation is: \[ y = 4x - 5 \] By comparing the two equations: \[ f(2) + f^{\prime}(2)(x - 2) = 4x - 5 \] 3. **Determine \( f^{\prime}(2) \):** Compare the coefficients of \( x \): \[ f^{\prime}(2) = 4 \] 4. **Determine \( f(2) \):** Compare the constant terms: \[ f(2) - 2f^{\prime}(2) = -5 \] Substitute \( f^{\prime}(2) = 4 \): \[ f(2) - 2(4) = -5 \implies f(2) = -5 + 8 \implies f(2) = 3 \] **Final Answer:** \[ f(2) = 3 \quad \text{and} \quad f^{\prime}(2) = 4 \]