Answer: x = 2.
- A x = 1
- B x = 2
- C x = 3
- D x = 4
Correct answer: B. x = 2
Explanation: Volume V = x(12-2x)<sup>2.</sup> dV/dx = (12-2x)<sup>2</sup> - 4x(12-2x) = (12-2x)(12-6x) = 0 gives x=6 (rejected, makes V=0) or x=2. Second derivative test confirms x=2 gives maximum volume.
The tangent at a point touches the curve with slope f'(a); the normal is the line perpendicular to the tangent at that same point, with slope -1/f'(a) - together they describe the curve's local direction and the line "straight into" the curve.
Concept context
Use derivatives to study rate of change, increasing and decreasing functions, tangents and normals, and maxima and minima, with classic optimization problems.