Find all extreme values (if any) of the given function on the given interval. Determine at which

step 1 Since since F -dash t equal to t plus 4 by t squared is greater than 0, 4, 4, 4 t is less than 1

Find all extreme values (if any) of the given function on...

Key Concepts

Behavior at Interval Boundaries

Evaluating the behavior of the function at the boundaries of the interval is essential, particularly for intervals that are not fully closed. This involves checking the function's limit approaching an open boundary and directly computing the function’s value at any included endpoints to ensure that all potential extreme values are considered.

Derivative Test

The derivative test involves finding where the derivative of a function equals zero or is undefined, and then analyzing the sign of the derivative around these critical points to determine whether these points correspond to local minima or maxima. This systematic approach helps in identifying potential extreme values.

Endpoints Consideration

When finding extreme values on an interval, especially a closed or half-open interval, it's crucial to consider the values at the endpoints along with the critical points. For intervals that are not fully closed, one must evaluate the function at the available endpoint and consider the behavior of the function as it approaches any endpoint that is not included.

Critical Points

Critical points are points in the interior of the domain where the derivative of the function is zero or undefined. These points are important because local minima or maxima can occur there, and they are key candidates when searching for extreme values.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it must attain both a maximum and a minimum value on that interval. However, if the interval is not closed (e.g., one endpoint is not included), extra care must be taken in evaluating the function's behavior at the boundaries.

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