Stationary Points & Turning Points

Stationary Point: A point on the graph of a function $y = f(x)$ where the first derivative, $f'(x)$ or $\frac{dy}{dx}$, is equal to zero. At such a point, the tangent line to the curve is horizontal, indicating a momentary halt in the function's rate of change.

Turning Point: A specific type of stationary point where the function changes its direction, meaning the derivative changes sign. Turning points are either local maximum points or local minimum points, representing peaks or troughs in the function's graph.

Relationship: All turning points are stationary points because their derivative is zero. However, not all stationary points are turning points; some stationary points, like points of inflection, have a zero derivative but do not change the function's direction.

Local Maximum: A point $(x_0, f(x_0))$ where $f(x_0)$ is the highest value of the function within a specific interval around $x_0$. The function increases before this point and decreases after it.

Local Minimum: A point $(x_0, f(x_0))$ where $f(x_0)$ is the lowest value of the function within a specific interval around $x_0$. The function decreases before this point and increases after it.

Gradient as Rate of Change: The first derivative, $f'(x)$, represents the instantaneous gradient of the tangent line to the curve $y = f(x)$ at any point $x$. This gradient indicates the rate at which the function's value is changing with respect to $x$.

Zero Gradient at Stationary Points: When $f'(x) = 0$, the gradient of the tangent line is zero, meaning the tangent is horizontal. This signifies that the function is momentarily neither increasing nor decreasing at that specific point, hence the term 'stationary'.

Sign Change for Turning Points: For a stationary point to be a turning point (a local maximum or minimum), the function's direction must change. This is reflected by a change in the sign of the first derivative: from positive to negative for a local maximum, and from negative to positive for a local minimum.

Concavity and the Second Derivative: The second derivative, $f''(x)$, provides information about the concavity of the function. A positive second derivative ($f''(x) > 0$) indicates concave up (like a cup), while a negative second derivative ($f''(x) < 0$) indicates concave down (like a frown). This property is crucial for the second derivative test.

Step 1: Differentiate the Function: Begin by finding the first derivative of the given function, $y = f(x)$, which is $f'(x)$ or $\frac{dy}{dx}$. This derivative function will describe the gradient of the curve at any point $x$.

Step 2: Set the Derivative to Zero: To locate stationary points, set the first derivative equal to zero: $f'(x) = 0$. Solving this equation will yield the x-coordinates of all stationary points, as these are the points where the gradient is horizontal.

Step 3: Find Corresponding y-coordinates: For each x-coordinate found in Step 2, substitute it back into the original function $y = f(x)$ to determine the corresponding y-coordinate. This gives the full coordinates $(x, y)$ of each stationary point.

Principle: This test determines the nature of a stationary point by examining the sign of the first derivative on either side of the point. It directly checks if the function is changing from increasing to decreasing or vice versa.

Methodology: After finding the x-coordinate of a stationary point, choose a test value slightly less than $x$ (e.g., $x - h$) and another slightly greater than $x$ (e.g., $x + h$), where $h$ is a small positive number. Evaluate $f'(x - h)$ and $f'(x + h)$.

Local Maximum: If $f'(x - h) > 0$ (increasing) and $f'(x + h) < 0$ (decreasing), the stationary point is a local maximum. The derivative changes from positive to negative.

Local Minimum: If $f'(x - h) < 0$ (decreasing) and $f'(x + h) > 0$ (increasing), the stationary point is a local minimum. The derivative changes from negative to positive.

Point of Inflection: If the sign of $f'(x)$ does not change across the stationary point (e.g., positive on both sides or negative on both sides), then it is a point of inflection, which is a stationary point but not a turning point.

Principle: This test uses the concavity of the function at a stationary point to classify its nature. The sign of the second derivative at the stationary point indicates whether the curve is concave up or concave down.

Methodology: First, find the second derivative of the function, $f''(x)$, by differentiating $f'(x)$. Then, substitute the x-coordinate of each stationary point into $f''(x)$.

Local Minimum: If $f''(x) > 0$ at the stationary point, the curve is concave up, indicating that the point is a local minimum. Think of a 'cup' shape holding the minimum.

Local Maximum: If $f''(x) < 0$ at the stationary point, the curve is concave down, indicating that the point is a local maximum. Think of a 'frown' shape peaking at the maximum.

Inconclusive Case: If $f''(x) = 0$ at the stationary point, the second derivative test is inconclusive. In this scenario, the first derivative test must be used to determine the nature of the stationary point, as it could be a local maximum, local minimum, or a point of inflection.

Justify Your Classification: Never assume that $f'(x)=0$ automatically means a local maximum or minimum. Always apply either the first or second derivative test to formally justify the nature of the stationary point. Examiners look for this justification.

Check for Inconclusive Cases: Be aware that the second derivative test can be inconclusive if $f''(x)=0$. In such cases, immediately switch to the first derivative test to correctly classify the stationary point. This is a common trap.

Calculate y-coordinates: Unless explicitly asked only for x-coordinates, always find the corresponding y-coordinates by substituting the x-values back into the original function $f(x)$. Providing full coordinates $(x,y)$ is crucial for a complete answer.

Small 'h' for First Derivative Test: When using the first derivative test, ensure your test values $x-h$ and $x+h$ are sufficiently close to the stationary point and do not 'jump over' any other stationary points or discontinuities. A small decimal value for $h$ (e.g., 0.1 or 0.01) is usually appropriate.

Calculator Use: While calculators can often find numerical derivatives at a point or graph functions, they typically cannot derive the symbolic derivative $f'(x)$. Use them to check your calculations or visualize the curve, but perform the differentiation steps manually.