Answer:
The rate of change of the function at x = 0.6 is approximately -0.72
Step-by-step explanation:
The given information from the graph are;
Points (0, 0) and (1, -0.5)
We have that the graph is that of a cubic function, therefore;
f(x) = ax³ + bx² + cx + d
At f'(x) at (0, 0) and (1, -0.5) = 0 gives;
3ax² + 2bx + c = 0
∴ c = 0
Also
ax³ + bx² + cx + d = 0
d = 0
We have
3a(1)² + 2b(1) + 0 = 0
3·a + 2·b = 0..............(1)
a(1)³ + b(1)² + c(1) + d = -0.5
a + b + 0×(1) + 0 = -0.5
a + b = -0.5..........(2)
Multiplying equation (2) by 2 and subtracting it from equation (1) gives;
2 × (a + b = -0.5) = 2·a + 2·b = -1
3·a + 2·b - (2·a + 2·b) = 0 - (-1) = 0 + 1 = 1
a = 1
From
a + b = -0.5, we have;
1 + b = -0.5 = 0
b = -0.5 - 1 = -1.5
The equation becomes
f(x) = x³ - 1.5·x²
The rate of change of the function at x = 0.6 is therefore given as follows;
f'(x) = 3 × x² - 2 × (1.5)·x = 3·x² - 3·x
At x = 0.6, we have;
f'(0.6) = 3·(0.6)² - 3·(0.6)
f'(0.6) = 3×0.6² - 3×0.6 = -0.72
The rate of change of the function at x = 0.6 ≈ -0.72
or 360, so D.