The above is a graph of and its first and second derivatives, to show how critical points, points of inflection, and maxima and minima are related among functions and their derivatives.
Here is some background on the relationship between functions and their first and second derivatives; the information is generalizable to any number of derivatives down the line:
Where a function is decreasing, its first derivative will be negative, i.e. below the x axis. Where the function is increasing, its first derivative will be positive, i.e. above the x axis. Where a function hits a plateau, say it’s increasing then hits a plateau then starts decreasing, or vice versa, that’s a maximum or minimum: where that happens, the first derivative will be zero, as it is either changing from positive to negative, or negative to positive, and thus has to cross the x axis.
Next we should look at the concavity of the function, and later learn about the second derivative test, otherwise known as the concavity test. Basically, if the function at a point is resting above the tangent line to that point, like a letter U, that’s concave up. If the function at a point is resting below the tangent line to that point, like an upside-down letter U, that’s concave down.
Where a function changes concavity is called a “point of inflection/point of inflexion.” So for example, think of a function may be increasing, and thus has a positive first derivative; where that increasing function changes concavity and is still increasing, the first derivative will still be positive, but will change direction, and thus have either a maximum or minimum at that point. On the graph above of , the function is decreasing but concave up, but then switches to concave down going from negative x and crossing zero to positive x. Since it is a decreasing function there, the first derivative is negative.
Where the concave was up, the first derivative was negative but increasing; then where the concavity switches to concave down and the function is still decreasing, the first derivative hits a maximum, then starts decreasing, and is still negative. The first derivative continues to be negative until the function switches concavity to concave up, hits a minimum, and starts increasing, which it does at x=3. Now, we know that where a function has a maximum or minimum, its first derivative will be zero. That’s because the tangent line at that maximum or minimum point is flat, horizontal; for any change in x there is no change in y because it is perfectly flat.
The second derivative test, otherwise known as the concavity test, is important: where a function is concave up, its second derivative is positive, and where a function is concave down, its second derivative is negative. If the second derivative is positive, meaning the function is concave up like a U, and the first derivative is zero, that means there is a minimum at that point on the function. If the second derivative is negative, meaning the function is concave down like an upside-down U, and the first derivative is a zero, that means there is a maximum at the point on the function. If the second derivative is equal to zero, and the first derivative is a zero, there may be a maximum, minimum, or point of inflection at that point on the function. See section 8.2 in the book Mathematics for Economists (read the first edition for free on Google Books!) for more information.
Now, where a function has a point of inflection, its first derivative will be a maximum or minimum, but also the value of the derivative may or may not be zero, i.e. it may or may not be a critical point, depending on if the tangent line at that point of inflection is horizontal or not. For more information on points of inflection being critical or non-critical, see footnote 2 on this page in the book Mathematics for Economists and the following information from Wikipedia:
Points of inflection can also be categorised according to whether f’(x) is zero or not zero.
- if f’(x) is not zero, the point is a non-stationary point of inflection
An example of a saddle point is the point (0,0) on the graph y=x³. The tangent is the x-axis, which cuts the graph at this point.
A non-stationary point of inflection can be visualised if the graph y=x³ is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero.
While we are looking at graphing, here are some other vital points to help understand graphing polynomials. (See Algebra II for Dummies for a quick, useful reference that brings together a lot of stuff your teacher may have neglected to cover, as usual, in your high school algebra/precalculus classes or that you simply didn’t pay attention to at the time. Lack of a thorough grasp of Algebra II probably accounts for 80% of the difficulty anyone ever has learning calculus, IMHO.)
First, an important thing to remember is that critical points are points that will equal zero in the function’s first derivative; zeros of the function itself are called roots, zeros, solutions, x-intercepts.
For polynomials, the maximum number of roots/zeroes/x-intercepts/solutions the polynomial can have is determined by the highest degree/the highest power of the polynomial: so for an equation like , there could be at most 8 x-intercepts.
Turning points are where the function has a maximum or minimum, changing directions from up to down or vice versa. A polynomial can have one less turning point than its highest degree, so in the equation , there could be at most 7 turning points.
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