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Stationary points on a graph are where the slope of the tangent to the curve is zero. This means that at these points, the curve neither rises nor falls for a tiny region around it. To find these points, we must first take the derivative of the function and set it equal to zero. The derivative tells us the slope of the tangent line at any given point. Therefore, by solving the equation resulting from setting the first derivative to zero, we can determine the x-values of stationary points. For example, if we have a function like \( y = x - \sqrt{x} \), finding the stationary point involves differentiating and solving \( \frac{dy}{dx} = 0 \). This helps us find where the turning points occur on the curve.

The first derivative of a function, often denoted as \( \frac{dy}{dx} \), indicates the rate at which the function's value changes with respect to changes in \( x \). It gives us the slope of the tangent line to the function's graph at any point. In simpler terms, the first derivative tells us how steep the graph is and in which direction it is moving. For a function like \( y = x - \sqrt{x} \), the first derivative is \( 1 - \frac{1}{2\sqrt{x}} \). By setting this equal to zero, we can solve for \( x \) to find the stationary points. Any value of \( x \) that makes the first derivative zero indicates a turning point of the function.

The second derivative of a function provides us with information about the curve's concavity; that is, it helps us understand whether the graph is curving upwards or downwards at a particular point. This is essential for determining the nature of stationary points. If the second derivative, noted as \( \frac{d^2y}{dx^2} \), is positive at a stationary point, the curve is concave up, which indicates a local minimum. Conversely, if the second derivative is negative, the curve is concave down, signifying a local maximum. In our earlier function \( y = x - \sqrt{x} \), using the second derivative \( \frac{1}{4\sqrt{x^3}} \) and substituting \( x = \frac{1}{4} \), we found it to be positive, confirming a minimum point at this stationary point.

Minimum and maximum points, also called extrema, refer to points on a graph where the function reaches its lowest or highest value, respectively, in a neighborhood. Finding these points involves using both the first and second derivatives. Once a stationary point is found using the first derivative, the second derivative test helps determine whether it is a minimum or maximum. A positive second derivative suggests a minimum, while a negative indicates a maximum. In the function \( y = x - \sqrt{x} \), our calculations showed a stationary point at \( \left( \frac{1}{4}, -\frac{1}{4} \right) \), with a positive second derivative, thus indicating a minimum.