According to the first derivative test, a differentiable function must have a relative minimum between any two relative maxima.
True.
when f' changes sign, that is max or min (first derivative test)
moving right on x axis
f' changes sign, goes - at left maximum
to get back up to right maximum, f' must change sign again from - to +
that is a minimum
To determine whether a differentiable function has a relative minimum between two relative maxima, we can use the first derivative test. Here's how we can apply the first derivative test:
1. Find the critical points of the function. These are the points where the derivative is either zero or undefined.
2. Determine the intervals between the critical points. Divide the number line into intervals based on the critical points. Each interval will have a different sign for the derivative.
3. Evaluate the sign of the derivative in each interval. Choose a test point from each interval and evaluate the derivative at that point. This will tell us whether the function is increasing or decreasing in that interval.
4. Identify the relative extrema. Look for changes in the sign of the derivative. A change from positive to negative indicates a relative maximum, while a change from negative to positive indicates a relative minimum.
Based on the first derivative test, if a differentiable function has two relative maxima, there must be a relative minimum between them. So the statement "According to the first derivative test, a differentiable function must have a relative minimum between any two relative maxima." is true.