Extreme Value Theorem EXIT

Front

What does the Extreme Value Theorem state?

Back

If a function is continuous on a closed interval, then it has a maximum and minimum value on that interval.

Front

What should you check when applying the Extreme Value Theorem?

Back

You should check the endpoints of the interval, as they may contain the maximum or minimum values.

Front

What is a critical point of a function?

Back

A critical point is where the derivative is zero or undefined, indicating potential local maxima or minima.

Front

How do you find critical points for a function?

Back

Set the derivative of the function equal to zero and solve for x.

Front

What is the significance of the second derivative in determining concavity?

Back

The second derivative indicates the concavity of the function: if it's positive, the function is concave up; if negative, concave down.

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x.

Front

How do you find horizontal tangent lines for a function?

Back

Set the first derivative equal to zero and solve for x.

Front

What does it mean for a function to be continuous?

Back

A function is continuous if there are no breaks, jumps, or holes in its graph over its domain.

Front

What is the relationship between critical points and local extrema?

Back

Critical points can indicate local extrema, but not all critical points are local maxima or minima.

Front

What is the formula for the second derivative of y with respect to x using implicit differentiation?

Back

The second derivative can be found by differentiating the first derivative with respect to x, applying the chain rule as necessary.