As the temperature is raised, the vapour pressure increases, and the gas phase becomes denser. The liquid expands and becomes less dense until, at the critical point, the densities of liquid and vapour become equal, eliminating the boundary between the two phases.

Important Points. The three phase equilibrium curves meet at the triple point. At the triple point, all three phases (solid, liquid, and gas) are in equilibrium. The critical point is the highest temperature and pressure at which a pure material can exist in vapor/liquid equilibrium.

In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.

Determine whether each of these critical points is the location of a maximum, minimum, or point of inflection. For each value, test an x-value slightly smaller and slightly larger than that x-value. If both are smaller than f(x), then it is a maximum. If both are larger than f(x), then it is a minimum.

Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.

If is a point where reaches a local maximum or minimum, and if the derivative of exists at , then the graph has a tangent line and the tangent line must be horizontal.

Extremum, plural Extrema, in calculus, any point at which the value of a function is largest (a maximum) or smallest (a minimum). There are both absolute and relative (or local) maxima and minima.

A relative maximum or minimum occurs at turning points on the curve where as the absolute minimum and maximum are the appropriate values over the entire domain of the function. In other words the absolute minimum and maximum are bounded by the domain of the function. So we have: Relative minimum of −9 occuring at x=1,3.

If a limit is infinity or negative infinity, these cannot be considered as the absolute extrema values. 3. The greatest function value is the absolute maximum value and the least is the absolute minimum value.

There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum. Assuming this function continues downwards to left or right: The Global Maximum is about 3.7. The Global Minimum is −Infinity.

On a closed interval, the justification of an absolute maximum or minimum can be accomplished by identifying all critical values as well as the endpoints, evaluating the function at each of these values, and then identifying which value of x corresponds to the absolute maximum or minimum of the function.

In order for a function f to have a relative maximum at a certain point, it must increase before that point and decrease after that point. At the maximum point itself, the function is neither increasing nor decreasing.

If f(x) is continuous on R and it has a local maximum at x0 and no other maximum or minimum points, then prove that x0 is a global maximum.

The slope of inflection point is not undefined, it can be any value, but its second derivative must be zero. The inflection point in tan(theta) occurs at theta = 0.

Explanation:

1. If a function is undefined at some value of x , there can be no inflection point.
2. However, concavity can change as we pass, left to right across an x values for which the function is undefined.
3. f(x)=1x is concave down for x<0 and concave up for x>0 .
4. The concavity changes “at” x=0 .

A sharp part on a derivative function will not form a cusp on the original function. That being said, there is no reason why we would not consider a function to have an inflection point at an x coordinate at which the function is not twice-differentiable.

If the function f(x) is continuous and differentiable at a point x0, has a second derivative f′′(x0) in some deleted δ-neighborhood of the point x0 and if the second derivative changes sign when passing through the point x0, then x0 is a point of inflection of the function f(x).

Inflection points (or points of inflection) are points where the graph of a function changes concavity (from ∪ to ∩ or vice versa).

Note: all turning points are stationary points, but not all stationary points are turning points. A point where the derivative of the function is zero but the derivative does not change sign is known as a point of inflection, or saddle point.