let f be the function given by f(x)=(4x^2)-x^3, and let l be the line y=18-3x, where l is tangent to the graph of f. Let D be the region bounded by the graph of f and the x-axis, and let E be the region bounded by the graph f, the line l, and the x-axis.

a. show that l is tangent to the graph of y=f(x) at the point x=3

Answer:

df/dx = 8x - 3x²

is the slope of the curve, which at x = 3 produces df/dx = 24 - 27 = -3

When x = 3, f(x) = 36 - 27 = 9.

The equation of a line passing through (3,9) with slope -3 is

y = -3x + b

9 = -3(3) + b

b = 18

→ y = 18 - 3x ← (a)

thats correct right?