To solve the problem, we need to analyze the given expression and its derivative step by step. ### Step 1: Understanding the Expression and Its Derivative The expression given is: \[ ax^2 + bx + c \] We know that at \( x = 1 \), this expression equals 5: \[ a(1)^2 + b(1) + c = 5 \] This simplifies to: \[ a + b + c = 5 \quad \text{(Equation 1)} \] The derivative of the expression is given as: \[ \frac{d}{dx}(ax^2 + bx + c) = 2ax + b \] We are told that this derivative equals \( 2x + 1 \). Therefore, we can set the two expressions equal to each other: \[ 2ax + b = 2x + 1 \] ### Step 2: Matching Coefficients To find the values of \( a \) and \( b \), we can match the coefficients of \( x \) from both sides of the equation. 1. **Coefficient of \( x \)**: - From \( 2ax \), the coefficient is \( 2a \). - From \( 2x \), the coefficient is \( 2 \). - Therefore, we have: \[ 2a = 2 \] Dividing both sides by 2 gives: \[ a = 1 \quad \text{(Equation 2)} \] 2. **Constant term**: - From \( b \), the constant term is \( b \). - From \( 1 \), the constant term is \( 1 \). - Therefore, we have: \[ b = 1 \quad \text{(Equation 3)} \] ### Step 3: Finding \( c \) Now that we have \( a \) and \( b \), we can substitute these values back into Equation 1 to find \( c \): Substituting \( a = 1 \) and \( b = 1 \) into Equation 1: \[ 1 + 1 + c = 5 \] This simplifies to: \[ 2 + c = 5 \] Subtracting 2 from both sides gives: \[ c = 3 \quad \text{(Equation 4)} \] ### Final Values Thus, we have found: - \( a = 1 \) - \( b = 1 \) - \( c = 3 \) ### Conclusion The values of \( a, b, c \) are \( 1, 1, 3 \) respectively. Therefore, the correct option is: **D. 1, 1, 3** ### Explanation of Other Options - **Option A: 1, 3, 1** - Here, \( a = 1 \), \( b = 3 \), and \( c = 1 \). Substituting these into Equation 1 gives \( 1 + 3 + 1 = 5 \), which is correct, but the derivative does not match \( 2x + 1 \) since \( b \) should be 1, not 3. - **Option B: 1, 2, 1** - Here, \( a = 1 \), \( b = 2 \), and \( c = 1 \). Substituting into Equation 1 gives \( 1 + 2 + 1 = 4 \), which does not satisfy the condition \( a + b + c = 5 \). - **Option C: 2, 1, 1** - Here, \( a = 2 \), \( b = 1 \), and \( c = 1 \). Substituting into Equation 1 gives \( 2 + 1 + 1 = 4 \), which does not satisfy the condition \( a + b + c = 5 \). ### Revision Summary - The expression \( ax^2 + bx + c \) equals 5 at \( x = 1 \) leads to the equation \( a + b + c = 5 \). - The derivative \( 2ax + b \) must equal \( 2x + 1 \), leading to \( 2a = 2 \) and \( b = 1 \). - Solving these equations gives \( a = 1 \), \( b = 1 \), and \( c = 3 \). - The correct answer is option D: \( 1, 1, 3 \).