To solve the problem, we need to find the value of \( k \) such that the gradient (or derivative) of the curve \( y = 2kx^2 + x + 1 \) at \( x = 1 \) equals 9. Let's break this down step-by-step. ### Step 1: Find the Derivative The first step is to differentiate the function \( y \) with respect to \( x \). The function is: \[ y = 2kx^2 + x + 1 \] To find the derivative \( \frac{dy}{dx} \), we apply the power rule of differentiation, which states that if \( y = ax^n \), then \( \frac{dy}{dx} = nax^{n-1} \). 1. Differentiate \( 2kx^2 \): - The derivative is \( 2k \cdot 2x^{2-1} = 4kx \). 2. Differentiate \( x \): - The derivative is \( 1 \). 3. Differentiate \( 1 \): - The derivative is \( 0 \) (since the derivative of a constant is zero). Putting it all together, we have: \[ \frac{dy}{dx} = 4kx + 1 \] ### Step 2: Evaluate the Derivative at \( x = 1 \) Next, we need to evaluate the derivative at \( x = 1 \): \[ \frac{dy}{dx} \bigg|_{x=1} = 4k(1) + 1 = 4k + 1 \] ### Step 3: Set the Derivative Equal to 9 According to the problem, the gradient at \( x = 1 \) is 9. Therefore, we set up the equation: \[ 4k + 1 = 9 \] ### Step 4: Solve for \( k \) Now, we solve for \( k \): 1. Subtract 1 from both sides: \[ 4k = 9 - 1 \] \[ 4k = 8 \] 2. Divide both sides by 4: \[ k = \frac{8}{4} = 2 \] ### Conclusion Thus, the value of \( k \) is \( 2 \). Therefore, the correct option is: **C. 2** ### Explanation of Other Options - **A. 4**: If \( k = 4 \), then \( 4k + 1 = 4(4) + 1 = 16 + 1 = 17 \), which is not equal to 9. - **B. 3**: If \( k = 3 \), then \( 4k + 1 = 4(3) + 1 = 12 + 1 = 13 \), which is also not equal to 9. - **D. 1**: If \( k = 1 \), then \( 4k + 1 = 4(1) + 1 = 4 + 1 = 5 \), which again is not equal to 9. ### Revision Summary - To find the gradient of a curve, differentiate the function with respect to \( x \). - Evaluate the derivative at the specified point to find the gradient at that point. - Set the evaluated derivative equal to the given gradient and solve for the unknown variable. - Always check other options to confirm they do not satisfy the condition given in the problem.