Consider the curve y=kxr, defined for x>0, that passes through the point (3,54). The line tangent to the curve at this point intersects the y-axis at (0,−6).
Parts:
(a) Using only the information provided, determine the slope of the tangent line at (3,54) and explain your reasoning.
(b) Express the derivative dxdy of y=kxr in terms of k, r, and x.
(c) By equating your result from part (a) with the derivative evaluated at x=3, derive an equation that relates k and r.
(d) Combine the relation from part (c) with the fact that the curve passes through (3,54) to solve simultaneously for k and r. Show all algebraic steps without skipping intermediate reasoning.
(e) Verify that the resulting function satisfies both original geometric constraints.