Question

Given that the point (4,10) is a max on the quadratic function, explain whether these statements are true or false:

a. The instantaneous rate of change at 4 is 0
b. f(6)>f(4)
d. There is only one possible value for x that has an instantaneous rate of change = 0.

Answer 1

(a) TRUE; (b) FALSE; (c) TRUE

Explanation:

a. The instantaneous rate of change at 4 is 0. TRUE.

The function is f(x) = a(x-4)^2 - 10; the derivative of this function is f '(x) = 2a(x-4), which must equal zero (0) at the vertex / max (4,10). Zero slope at the vertex corresponds with instantaneous rate of change zero at that point.

b. f(6)>f(4). FALSE.

Using the function given in (a), above, f(x) = a(x-4)^2 - 10. Since (4,10) is the max of this function, f is increasing on (-infinity, 4) and decreasing on (4, infinity). Thus, f(6) is smaller than f(4).

c. There is only one possible value for x that has an instantaneous rate of change = 0. TRUE

The inst. rate of change takes on the value 0 only at (4,10), since (4,10) is the vertex of this downward-opening parabolic graph.