Consider the following for the two (02) items that follow: The slope of the tangent to the curve y = f(x) at (x, f(x)) is 4 for every real number x and the curve passes through the origin. What is the nature of the curve?

Step-by-step solution

Given that the slope of the tangent to the curve y = f(x) is 4 for all x . dy dx = 4 Integrating both sides with respect to x : y = 4x + C Since the curve passes through the origin (0, 0) , substituting x = 0 and y = 0 gives: 0 = 4(0) + C C = 0 The equation of the curve is y = 4x . This represents a straight line. Checking the given points, for x = 1 , y = 4(1) = 4 . Thus, the straight line passes through the point (1, 4) .