Critical points

Front

What are critical points in a function?

Back

Critical points are values of x in a function where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection.

Front

How do you find critical numbers of a function?

Back

To find critical numbers, take the derivative of the function, set it equal to zero, and solve for x. Also, check where the derivative does not exist.

Front

What is the derivative of f(x) = x^2 + 2x + 1?

Back

The derivative is f'(x) = 2x + 2.

Front

What is the significance of a relative maximum?

Back

A relative maximum is the highest point in a particular section of a graph, indicating that the function value is greater than the values of the function at nearby points.

Front

What is the derivative of g(x) = 2x^3 - 3x^2?

Back

The derivative is g'(x) = 6x^2 - 6x.

Front

How do you determine if a critical point is a maximum or minimum?

Back

Use the first derivative test: if the derivative changes from positive to negative at a critical point, it's a maximum; if it changes from negative to positive, it's a minimum.

Front

What is the derivative of ax, where a is a constant?

Back

The derivative is a.

Front

What is the second derivative test?

Back

The second derivative test uses the second derivative of a function to determine the concavity at critical points: if f''(x) > 0, it's a local minimum; if f''(x) < 0, it's a local maximum.

Front

Find the critical points of f(x) = x^3 + x^2 + 3.

Back

First, find the derivative f'(x) = 3x^2 + 2x, then set it to zero: 3x^2 + 2x = 0, giving critical points at x = 0 and x = -2/3.

Front

What is the relative minimum value?

Back

A relative minimum value is the lowest point in a particular section of a graph, indicating that the function value is less than the values of the function at nearby points.