Question

Let f(x)=x³-4. Which of these is the equation for the normal line to this curve at the point (2,4)?

A. y=1/12x-25/6

B. y=12x-25/6

C. y=-12x-4

D. y=-1/12x-2

E. y=-1/12x+25/6

Answer 1

The correct equation for the normal line to the curve at the point (2, 4) is y = -1/12x + 25/6, which is option E.

Step-by-step explanation:

The question deals with finding the equation of the normal line to the curve f(x) = x³ - 4 at the point (2, 4). To find this, we first need the slope of the tangent to the curve at x=2. This is the derivative f'(x) at x=2.

The derivative f'(x) = 3x². Therefore, at x=2, the slope of the tangent is f'(2) = 3(2)² = 12. The slope of the normal line is the negative reciprocal of the slope of the tangent, so it is -1/12.

The normal line passes through (2, 4), so its equation will be in the form y = mx + b. Using the point-slope form, we plug in our values: y - 4 = (-1/12)(x - 2). Simplifying this, we get y = -1/12x + 25/6. This corresponds to option E.