- 17 Stationary Points of Functions A stationary point of a curve is a point on the curve at which the gradient is zero. In this section we will focus on the definitions of different types of stationary points of functions and how to find them. By the end of this section, you should have the following skills: • An understanding of the definition of types of stationary points, in- cluding: local maxima, local minima and points of inflection. • Find stationary points of a function using differentiation, including: local maxima, local minima and points of inflection. • Test points to see if they are a stationary point. • Test stationary points to determine their type of stationary point, using first and second derivatives. 17.1 Local Maximum and Local Minimum Let f(x) be a function. It is useful to be able to find and classify local maxima and minima. These are points at which the function switches from increasing to decreasing and vice-versa. 17.1.1 Local Minimum Here the graph is decreasing and then increasing as x increases through the local minimum. The point x = b in the diagram is a local minimum. 17.1.2 Local Maximum In contrast, the graph is increasing and then decreasing as x increases through the local maximum. The point x = a in the diagram is a local maximum. 1 Local minimum and local maximum. Note that such a point as a is only a local maximum as it is possible that the function takes values elsewhere which are greater than the value at a. In our example, there are clearly values for the function which are greater than the value at x = a. Similarly for a local minimum. Example 1 1. f(x) = 1 has a local maximum and a local minimum at every point! 2. Any linear function f(x) = ax+b; a 6= 0; b constants has neither local maxima nor local minima. 3. f(x) = x2 has a local minimum at x = 0. 4. f(x) = −x2 has a local maximum at x = 0. 5. a quadratic has either one local minimum or one local maximum (but not both). 6. f(x) = x3 − 3x has a local maximum at x = −1 and a local mini- mum at x = 1. 2 7. f(x) = sin(x) has local maxima at x = π=2; 5π=2 9π=2;::: and has local minima at x = 3π=2; 7π=2 11π=2;:::. 8. f(x) = tan(x) has no local maxima or minima. 9. f(x) = ex has no local maxima or minima. 10. f(x) = ln(x) has no local maxima or minima. 11. f(x) = 1=x has no local maxima or minima. 12. f(x) = ex − x has a local minimum at x = 0 where f(0) = 1. Note that this shows that ex − x ≥ 1 ) ex ≥ 1 + x for all x. 3 13. 1 f(x) = (x − 1)(x − 2) has a local maximum at x = 3=2 where f(3=2) = −4. Note carefully that a local maximum is not a global maximum in general. For example f(x) = x3 −3x has a local maximum of f(−1) = 2 at x = −1 but this is not a maximum for the function as clearly there are points at which this function is greater than 2. What is important here is that all other points x sufficiently near to −1 have f(x) ≤ 2. Similarly for local minima, 4 x = 1 is a local minimum for f(x) = x3 − 3x but not a global minimum. Exercise 1 Each of the following functions f(x) has a local maximum or mini- mum as indicated. Which of these is a global maximum or minimum? (a) f(x) = x2 has a local minimum at x = 0. (b) f(x) = −x2 has a local maximum at x = 0. (c) f(x) = sin(x) has a local maximum at x = π=2. (d) f(x) = sin(x) has a local minimum at x = 3π=2. (e) f(x) = 1=(x − 1)(x − 2) has a local maximum at x = 3=2. (f) f(x) = ex − x has a local minimum at x = 0. Solutions to exercise 1 (a) f(x) = x2 has a local minimum at x = 0. (b) f(x) = −x2 has a local maximum at x = 0. 5 Graph of x2 and −x2. Clearly y = x2 has a global maximum at x = 0 whereas y = −x2 has a local minimum at x = 0. (c) f(x) = sin(x) has a local maximum at x = π=2. (d) f(x) = sin(x) has a local minimum at x = 3π=2. Since sin(π=2) = 1 and sin(3π=2) = −1 and for all x, −1 ≤ sin(x) ≤ 1 we see that x = π=2 is a global maximum and x = 3π=2 is a global minimum. (e) f(x) = 1=((x − 1)(x − 2)) has a local maximum at x = 3=2. The vertical lines at x = 1 and x = 2 are asymptotes. 6 We see that x = 3=2 is not a global maximum as for example f(0) = 1=2 > f(3=2): (f) f(x) = ex − x has a local minimum at x = 0. Note that if y = ex−x then dy=dx = ex−1 and so for x > 0; dy=dx > 0 and for x < 0; dy=dx < 0, hence the function is decreasing for x < 0 and increasing for x > 0, so x = 0 must be a global minimum. 7 17.2 Finding Local Maxima and Minima 17.2.1 Reminder about increasing and decreasing functions The following results from Increasing and Decreasing Functions are used in this section. Note that these results are true if f(x) is differentiable and f 0(x) is continuous, which we assume. 1. If f(x) is increasing for a range of values of x then df ≥ 0 dx over that range of values of x. 2. If f(x) is decreasing for a range of values of x then df ≤ 0 dx over that range of values of x. 17.2.2 Local Maxima We deal first with local maxima - the details for local minima are similar. First recall that a local maximum at x = a occurs when a function changes from increasing to decreasing i.e. • in an interval before b ≤ x ≤ a, f(x) is increasing ) df=dx ≥ 0; b ≤ x ≤ a and • in an interval after a ≤ x ≤ c, f(x) is decreasing ) df=dx ≤ 0; a ≤ x ≤ c. 8 Local Maximum. Since df=dx ≥ 0; df=dx ≤ 0; x = a we must have: If x = a is a local maximum then df = 0; x = a dx and df=dx decreases from positive to negative as x goes through a. 17.2.3 Local Minima Local Minimum. In a similar way we can show that if x = a is a local minimum then df = 0; x = a dx and df=dx increases from negative to positive as x goes through a. 17.2.4 Summary We have shown that if x = a is a local maximum or a local minimum then df = 0; x = a: dx This tells us that the tangent at x = a is horizontal i.e. parallel to the x-axis. 9 17.3 Stationary Points Any point x = a such that df = 0; x = a dx is called a stationary point. This is because the rate of change at x = a is 0 i.e. instantaneously stationary. So a local maximum or minimum is a stationary point. Note that for a local maximum or a local minimum the sign of df=dx changes as x goes through the stationary point. 17.3.1 Points of inflection There is another type of stationary point called a point of inflection. At these points we still have df=dx = 0 but df=dx does not change sign as x goes through a. For example y = x3 has a stationary point at x = 0, but dy=dx = 3x2 ≥ 0 for all x. So dy=dx does not change sign. Similarly, y = −x3 has a point of inflection at x = 0 as dy=dx ≤ 0 for all x. Figure 1: y = x3. 10 17.4 Finding Stationary Points If f(x) is a function we solve the equation df = 0 dx for x. The values of x we find are the stationary points of f(x). Example 2 Find the stationary points for the following functions f(x): (a) f(x) = x2 − 5x + 6. (b) f(x) = x3 − 6x2 + 7. (c) f(x) = x3 + 3x + 5. (d) f(x) = sin(x). (e) f(x) = ex − 2x. (f) f(x) = (x − 1)=(x2 + 5x + 3). (g) f(x) = ln(x3 − 6x2 − 15x + 1). (h) f(x) = 3x5 − 5x3 + 4. p (i) f(x) = 7 + (2x2 − 10x) x. Solution. (a) df = 2x − 5: dx The stationary points are given by x such that df 5 = 0 ) x = : dx 2 So one stationary point x = 5=2. 11 Graph of x2 − 5x + 6. (b) df = 3x2 − 12x: dx The stationary points are given by x such that df = 0 ) 3x2 − 12x = 3x(x − 4) = 0 ) x = 0; x = 4: dx So the stationary points are x = 0; 4. Graph of x3 − 6x2 + 7. 12 (c) df = 3x2 + 3: dx The stationary points are given by x such that df = 0 ) 3x2 + 3 = 0 ) x2 = −1: dx But there are no real number solutions of this equation. So no sta- tionary points. (d) df = cos(x): dx The stationary points are given by x such that df = 0 ) cos(x) = 0 ) x = ±π=2; ±3π=2; ±5π=2;:::: dx There are an infinite number of stationary points.