Federal Deficit The deficit of the federal government (in...

Federal Deficit The deficit of the federal government (in trillions of dollars) in year $t$ from $2006(t=0)$ through $2012(t=6)$ is approximately $$D(t)=-0.038898 t^{3}+0.30858 t^{2}-0.31849 t+0.22 \quad(0 \leq t \leq 6)$$ What was the largest federal deficit over the period under consideration? Hint: Use the quadratic formula.

Federal Deficit The deficit of the federal government (in trillions of dollars) in year $t$ from $2006(t=0)$ through $2012(t=6)$ is approximately
$$D(t)=-0.038898 t^{3}+0.30858 t^{2}-0.31849 t+0.22 \quad(0 \leq t \leq 6)$$
What was the largest federal deficit over the period under consideration?
Hint: Use the quadratic formula.

Key Concepts

Global Maximum Analysis on a Closed Interval

When optimizing a function over a closed interval, it is important to evaluate both the critical points found by setting the derivative to zero and the endpoints of the interval. This process ensures that the absolute maximum or minimum value of the function is correctly identified.

Quadratic Formula

The quadratic formula is a method for finding the roots of a quadratic equation. When the derivative of a polynomial function is quadratic, the quadratic formula can be used to solve for the critical points where the derivative equals zero.

Derivatives

Derivatives measure the rate at which a function's value changes as its input changes. In the context of optimization, taking the derivative of a function helps determine where the function's slope is zero, indicating potential local minima or maxima.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of a variable multiplied by coefficients. They often model real-life situations and allow for the analysis of behavior such as growth, decay, or turning points over an interval.

Critical Points and Optimization

Critical points occur where the derivative of a function is zero or undefined, representing potential locations for local maximum or minimum values. Identifying these points is essential in optimization problems to find the largest or smallest values of a function over a given interval.

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