Using the Mean Value Theorem

Mean value theorem states that if f is continuous on the closed interval and differentiable on the open interval, there is at least one point where the instantaneous rate of change is equal to the average rate of change

Extreme Value Theorem

Extreme value theorem states that if f is continuous on the closed interval, then it has an absolute maximum and an absolute minimum on the interval

There are two places where an extreme value can occur -- a critical point or an endpoint

Compare the local extrema to find the global extrema

Intervals on Which a Function is Increasing/Decreasing

Critical values are when the derivative is 0 or the derivative doesn’t exist

If the sign changes from positive to negative, it’s a relative maximum

If the sign changes from negative to positive, it’s a relative minimum

First Derivative Test to Determine Local Extrema

First derivative test is when you plug intermediate values between the critical points into the derivative to test for a sign change

Determining Concavity

Concave up is when the rate of change over an interval is increasing (getting more steep)

Concave down is when the rate of change of over an interval is decreasing (getting less steep)

Second Derivative Test to Determine Extrema

If the second derivative is positive, then the function is concave up at that point

If the second derivative is negative, then the function is concave down at that point

Points of inflection are where the function switches concavity (the second derivative of this point will be 0)

If a local extrema is concave up, it is a minimum

If a local extrema is concave down, it is a maximum

Optimization

Find a relationship between two variables to get rid of one

Then find the derivative

Then find the local minimum/maximum based on the question