Question

\(y=2 x^{2}-x+1\) \(\frac{d y}{d x}=4 x_{1}-1\) \(4 x_{1}-1=3\) \(x_{1}=1\) \(y_{1}=2\)

Short answer

Answer: The point (x1, y1) on the curve is (1, 2).

Step-by-step solution

Find the derivative of the function

To find the derivative of the given function, \(y = 2x^2 - x + 1\), with respect to x, we can apply the power rule, which states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). So the derivative is: $$\frac{dy}{dx} = 2(2x^{2-1}) - (1 \cdot x^{1-1}) = 4x - 1$$

Solve the equation of the derivative

Now, we know that the derivative is given by the equation \(4x_1 - 1 = 3\). In order to find the value of \(x_1\), we have to solve this equation: $$4x_1 - 1 = 3 \Rightarrow 4x_1 = 4 \Rightarrow x_1 = 1$$

Find the value of y1

To find the corresponding y1 value for x1=1, we have to plug this value into the original function: $$y_1 = 2(1)^2 - (1) + 1 = 2 -1 + 1 =2$$

Write the final answer

Now that we have found the x1 and y1 values, we can write the final answer as a point on the curve: The point is \((x_1, y_1) = (1, 2)\).