Curve sketching review

How can you determine if a critical point is a local extremum using the second derivative test?

• If f''(x) > 0, the point is a local minimum; if f''(x) < 0, the point is a local maximum.

• If f''(x) = 0, the point is a local extremum.

• If f'(x) > 0, the point is a local maximum; if f'(x) < 0, the point is a local minimum.

• If f''(x) > 0, the point is a local maximum; if f''(x) < 0, the point is a local minimum.

For a function g(x), g''(3)=-8 indicates that g(x) is ____________ at x=3.

• concave up

• linear

• concave down

• increasing

What does it mean if the second derivative of a function is zero at a point?

• It indicates a potential inflection point where the concavity might change.

• It means the function has a local maximum at that point.

• It suggests that the function is increasing at that point.

• It indicates that the function is constant at that point.

What does the first derivative test help determine?

• It helps determine local maxima and minima of a function.

• It helps find the global maximum of a function.

• It helps calculate the area under a curve.

• It helps identify points of inflection.

What is the purpose of sketching the derivative of a function?

• To find the roots of the function.

• To analyze the behavior of the original function, including increasing/decreasing intervals and critical points.

• To calculate the area under the curve.

• To determine the maximum value of the function.

Which graph is the derivative of the top graph?

• A

• B

• C

• D

How do you find the x-intercepts of a function?

• Set f(x) = 0 and solve for x.

• Evaluate f(x) at x = 0.

• Find the derivative of f(x).

• Graph the function and look for intersections with the x-axis.

What does it mean if the first derivative of a function is zero at a point?

• The function has a critical point at that location.

• The function is increasing at that point.

• The function is decreasing at that point.

• The function has a local maximum at that location.

What does it mean if the second derivative of a function is positive at a point?

• The function is concave up at that point.

• The function is concave down at that point.

• The function has a local maximum at that point.

• The function is linear at that point.

For a function f(x), f'(-3) = 5 indicates f(x) is ___________ at x=-3.

• decreasing

• constant

• increasing

• undefined