as saddle points. By Fermat's theorem, global extrema must occur (for a function) on the boundary or at stationary points. Curve sketching Determining the position and nature of stationary points aids in curve sketching of differentiable functions. Solving the equation f'(x) = 0 returns the x-coordinates of all stationary points; the y-coordinates are trivially the function values at those x-coordinates. The specific nature of a stationary point at x can in some cases be determined by examining the second derivative f''(x): If f''(x) < 0, the stationary point at x is concave down; a maximal extremum. If f''(x) > 0, the stationary point at x is concave up; a minimal extremum. If f''(x) = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point. A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points (if the function is defined and continuous between them). A simple example of a point of inflection is the function f(x) = x3. There is a clear change of concavity about the point x = 0, and we can prove this by means of calculus. The second derivative of f is the everywhere-continuous 6x, and at x = 0, f′′ = 0, and the sign changes about this point. So x = 0 is a point of inflection. More generally, the stationary points of a real valued function are those points x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero. Example For the function f(x) = x4 we have f'(0) = 0 and f''(0) = 0. Even though f''(0) = 0, this point is not a point of inflection. The reason is that the sign of f'''(x) changes from negative to positive. For the function f(x) = sin(x) we have f'(0) ≠ 0 and f''(0) = 0. But this is not a stationary point, rather it is a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of f'(x) does not change; it stays positive. For the function f(x) = x3 we have f'(0) = 0 and f''(0) = 0. This is both a stationary point and a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the