To find the relative extreme points of the function $T(t) = -0.2t^2 + 2.4t + 98.6$, we need to follow these steps:

1. Find the derivative of the function $T(t)$.
2. Set the derivative to zero to find the critical points.
3. Determine whether each critical point is a maximum or minimum by using the second derivative test or by evaluating the sign changes of the first derivative.

Step 1: Find the derivative of $T(t)$

The first derivative $T'(t)$ is:

$T'(t) = \frac{d}{dt} \left( -0.2t^2 + 2.4t + 98.6 \right) = -0.4t + 2.4$

Step 2: Set the derivative equal to zero

Set $T'(t) = 0$:

$-0.4t + 2.4 = 0$

Solving for $t$:

$-0.4t = -2.4 \implies t = 6$

Step 3: Determine the nature of the critical point

To classify whether $t = 6$ is a maximum or minimum, we can use the second derivative test. The second derivative $T''(t)$ is:

$T''(t) = \frac{d}{dt} (-0.4t + 2.4) = -0.4$

Since $T''(t) = -0.4 < 0$, this indicates that $T(t)$ has a relative maximum at $t = 6$.

Relative Extreme Points

The relative maximum occurs at $t = 6$. To find the corresponding temperature $T(6)$:

$T(6) = -0.2(6)^2 + 2.4(6) + 98.6$ $T(6) = -0.2(36) + 14.4 + 98.6$ $T(6) = -7.2 + 14.4 + 98.6 = 105.8$

So, the relative maximum point is $(6, 105.8)$.

Graph of the Function

To sketch the graph of $T(t)$ over the interval $0 \leq t \leq 12$, note the following key points:

• The relative maximum occurs at $t = 6$ with $T(6) = 105.8$.
• The endpoints $T(0) = 98.6$ and $T(12) = 98.6$ can be calculated directly.

Here’s a quick sketch of the graph showing the relative maximum at $(6, 105.8)$:

   T(t)
    |
106 |                *
    |              / \
    |             /   \
    |            /     \
    |           /       \
    |          /         \
 99 |_________/           \________
    |_________________________________ t
    0         6                        12

This sketch shows the temperature peaking at 105.8°F on the sixth day and being lower at both the start and end of the period.

Do you want more details or have any questions?

Related Questions

1. How do you find inflection points of a function?
2. How does the sign of the second derivative affect the shape of the graph?
3. What are the implications of a negative second derivative for a real-world temperature model?
4. Can there be more than one relative maximum in a quadratic function?
5. How would changing the coefficients of the function affect the location of the maximum?

Tip

To better understand the behavior of the function, always analyze both the first and second derivatives for critical points and concavity.