Secant and Tangent Lines

Question

f(x) = {3x, x ≥ 0

-2x, x < 0

Let g_d(x) be the average rate of change for f(x) over the interval [x, x+d], where d is positive fixed value. What is the point x* for which the absolute value of g_d(x) becomes the smallest?

Answer 1

The average rate of change is [f(x+d)-f(x)]/(x+d -x) = [f(x+d)-f(x)]/d

Consider | [f(x+d)-f(x)]/d |. When is this the smallest? Consider 3 cases.

Case 1: Both x and x+d are both negative. So compute | [f(x+d)-f(x)]/d |

Case 2: x<0 and x+d >0. [f(x+d)-f(x)]/d. So compute | [f(x+d)-f(x)]/d |

Case 3. Both x and x+d are positive. So compute | [f(x+d)-f(x)]/d |