Stationary Points Definition – Stationary Point • A stationary point of a function f, is the point (c, f(c)), where c is in the domain of f, such that f’(c) = 0. • i.e. for y = f(x), dy 0 dx Stationary points • Note that a stationary point is a coordinate pair (x, y) and all stationary points are critical points. Example 1 Find the stationary points of the function f(x) = 4x3 – 3x2 + 9. So, we must differentiate and find values of x for which f’(x) = 0. Example 1 f x 4 x 3 3x 2 9 f ' x 0 x0 f ' x 12 x 2 6 x 12 x 2 6 x 0 or x 0 f x 9 6 x2 x 1 0 1 x 2 and 1 35 x f x 2 4 Hence, the stationary points are: 0,9 and 1 35 , 2 4 Determining the nature of a stationary point • If f’(x) > 0 for all x within an interval then the function is increasing for that interval. f(x) f(x) x f(x) x x Determining the nature of a stationary point • If f’(x) < 0 for all x within an interval then the function is decreasing for that interval. f(x) f(x) x f(x) x x Determining the nature of a maximum stationary point • Consider a maximum stationary point: Maximum Determining the nature of a maximum stationary point • To the left of the maximum stationary point the function is increasing and to the right the function is decreasing. • Therefore, by the definition of a increasing and decreasing function, the derivative has changed sign from positive to negative. f’(x) > 0 f’(x) < 0 Maximum Determining the nature of a minimum stationary point • Consider a minimum stationary point: Minimum Determining the nature of a minimum stationary point • To the left of the minimum stationary point the function is decreasing and to the right the function is increasing. • Therefore, by the definition of a increasing and decreasing function, the derivative has changed sign from negative to positive. Minimum f’(x) < 0 f’(x) > 0 First Derivative Test • This process is known as the first derivative test. • If the sign of f’(x) changes from positive to negative either side of the stationary point then it is a maximum point. • If the sign of f’(x) changes from negative to positive either side of the stationary point then it is a minimum point. Example 6 Find the stationary points of y = 2x3 – 6x + 2 and determine their nature using the first derivative test. Solution: Differentiate and find values of x for which y’ = 0 and then evaluate y’ either side of the stationary point(s) to determine the nature. Example 6 y 2x 6x 2 3 y' 6 x 6 2 y' 0 6 x 6 0 6 x 1 0 2 x 1 0 2 x 1 y 2 and x 1 y 6 x 1 2 Example 6 We must now determine the nature of the stationary points using the first derivative test: x -2 -1 0 1 2 y’ = 6x2 – 6 18 0 -12 0 18 Either side of x = -1, y’ changes from positive to negative. Hence, (-1, 2) is a maximum point. Either side of x = 1, y’ changes from negative to positive. Hence, (1, 6) is a minimum point. First Derivative Test • We have considered the first derivative test when the gradient changes from positive to negative or from negative to positive. • Is it possible that the gradient does not change on either side of a stationary point? • To answer this we will consider y = x3. y = x3 • We know that y = x3 looks like this: y’ = 0: y’ = 3x2 = 0 obtains x = 0. Therefore (0, 0) is a stationary point but it is clearly not a maximum nor a minimum. The first derivative test confirms that the stationary point at x = 0 is neither a maximum nor a minimum since y’ > 0 on either side of the stationary point. So what is it? Stationary points of inflexion • This type of stationary point is called a stationary point of inflexion. First Derivative Test • So, we have 3 outcomes from the first derivative test. • If the sign of f’(x) changes from positive to negative either side of the stationary point then it is a maximum point. • If the sign of f’(x) changes from negative to positive either side of the stationary point then it is a minimum point. • If the sign of f’(x) is the same on either side of the stationary point then it is a stationary point of inflexion. Concavity and the second derivative • If the tangent to the curve is positioned below the curve then the curve is called concave . y’ < 0 y’ > 0 y’ = 0 • if y” > 0 then the curve is concave . Concavity and the second derivative • If the tangent to the curve is positioned above the curve then the curve is called convex. y’ = 0 y’ > 0 y’ < 0 • if y” < 0 then the curve is convex