We can find the stationary points of a function $f(x)$ using the following method:ġ) Find the (first) derivative of the function with respect to $x$. Note:The maximum and minimum points of a function are also called turning points because the slope of the function turns from being positive to being negative. They are called stationary points because they are points where the function is neither increasing nor decreasing. At a stationary point the slope of the graph of the function is zero (a straight line). Stationary PointsĪ stationary point of a function is a point where the derivative of a function is equal to zero and can be a minimum, maximum, or a point of inflection. We will cover the basic notation, relationship between the trig functions, the right triangle definition of the trig functions. Maximum and minimum points of a function are collectively known as stationary points. In this section we will give a quick review of trig functions. Now the binomial expansion is only defined for integers, therefore this proof is only for integer exponents of x.In economics, it is often important to know when the output of a function is at its highest or lowest possible value (maximum and minimum respectively).įor example, a firm may want to know how much it needs to sell to maximize its profit function or minimize its cost function (see the worked examples below.įinding the maximum and minimum points of a function requires differentiation and is known as optimisation. Things You Should Know An inflection point is where a function changes concavity and where the second derivative of the function changes signs. Now each term of the denominator contains an h but all contain higher powers of h than 1 except for the first term, therefore when we divide each term by h and take the limit as h → 0, all terms but the first vanish, leaving Here is the derivative expression for this function it's just a matter of plugging the function into the difference quotient expression: ![]() The function (see graph below right) has a positive slope for all x in its domain, except for x = 0, at which the slope is undefined. Therefore we have a test to determine if an interval is increasing or decreasing. ![]() It has a vertical asymptote at x = 1, therefore finding the slope of f(x) at x = 1 makes no sense – the function has no actual value there. ![]() This is an interesting function–a rational function. We'll wait for a while to actually prove that this limit is the slope of f(x) at x. The graphs below will help you visualize how making the distance between our two points smaller gives us a better and better estimate of the slope of f(x) at x. That is the essence of differential calculus: Finding the limit of a slope function as the change in the independent variable approaches zero. While we can all visualize the minimum and maximum values of a function we want to be a little more specific in our work here. Many of our applications in this chapter will revolve around minimum and maximum values of a function. Now do the following thought exercise: Imagine that Δx gets smaller and smaller, and eventually vanishes or becomes "infinitessimally small." In the limit that Δx = 0, we would have the exact slope of our function at point x. Section 4.3 : Minimum and Maximum Values. Convince yourself that this equation still just represents rise over run, Δy / Δx (and don't forget that y = f(x)). The slope of our secant line ( magenta), m, is written above the graph in terms of x, Δx, f(x) and f(x+Δx). ![]() This will allow us to reduce the width of the interval between them (the x-distance) by reducing Δx. To do that, we'll first change our notation a little and use x and ( x + Δx) instead of x 1 and x 2 to label our two points. This time we'll try to find the slope of our curve at some specific point x.
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