Final answer:
To find the value of c for tangency between the line y = 1/4x + 1 and the curve y = cx, we equate the slope of the line (0.25) to the derivative of the curve (c), which gives us c = 0.25.
Step-by-step explanation:
To find the value of c such that the line y = ⅔x + 1 is tangent to the curve y = cx, we need to look for the condition where these two equations have exactly one point of intersection. This means the slope of the line (⅔ or 0.25) must equal the slope of the curve (which is the derivative of the curve, or c in y = cx). Therefore, equating the slope of the line to the derivative of the curve, we get c = 0.25.
To confirm that this is indeed a tangential intersection and not a crossing, we need to compare the y-coordinates at the point where x is the same for both equations. If both y-coordinates are equal at this x value and c is established as 0.25, we can be sure that it's a tangential point and not a crossing point.