An analyst predicts that the demand for homeowner’s insurance will continue to decline for a time, and then begin to climb. Her model for the number of insurance policies demanded, shown in the following graph, is D = 0.0625t2 − 0.75t + 4.25, where D represents tens of millions of policies and t is time in months. On what interval does the analyst predict that the demand for policies will increase? A.
t ≥ 6

B.
4 ≤ t ≤ 10

C.
t ≥ 8

D.
0 ≤ t ≤ 6

option A. t ≥ 6

Explanation:

The function D = 0.0625t² − 0.75t + 4.25  is a quadratic function; thus, its graph is a parabola.

Since the coefficient of the quadratic term is positive, the parabola opens upward and the vertex is the minimum point of the curve.

That means that the curve decreases, reachs the minimum and then increases.

Then, the demand will increase from the time equal to the x-coordinate (t) of the of the vertex onwards.

Determine the vertex:

The x-coordinate of the vertex of a parabola given by the general form y = ax² + bx + c is equal to -b/(2a).

For D = 0.0625t² − 0.75t + 4.25, b = -0.75 and a = 0.0625, then:

x-coordinate of the vertex = - (-0.75)/(2 × 0.0625) = 6.

Therefore, the demand will increase from the month 6 onwards. That is t ≥ 6 (option A.).

answer: 48

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