The gradient of point P on the curve \(y = 3x^{2} - x + 3\) is 5. Find the coordinates of P.

To find the coordinates of point P, we need to differentiate the given equation with respect to x to get the derivative of the function. The derivative of y with respect to x gives us the gradient or slope of the tangent at any point on the curve. Taking the derivative of y with respect to x, we get: \[\frac{dy}{dx} = 6x - 1\] We are given that the gradient at point P is 5. So, we can equate the derivative of y with 5 and solve for x. \[6x - 1 = 5\] \[6x = 6\] \[x = 1\] Now that we know x = 1, we can find the corresponding y-coordinate by substituting x=1 into the original equation for y. \[y = 3x^{2} - x + 3\] \[y = 3(1)^{2} - 1 + 3\] \[y = 5\] Therefore, the coordinates of point P are (1, 5). To summarize, we found the x-coordinate of point P by equating the derivative of the equation with the given gradient of 5 and solved for x. Then, we found the corresponding y-coordinate by substituting the value of x into the original equation.