wat is different between critical point and inflection point?

Answer 1

There are three kinds of critical point -- a maximum, a minimum, or an inflection point.

These correspond respectively to the second derivative being negative, positive, or 0.

Alternatively, you can say they correspond respectively to the curve being concave downwards, concave upwards, or changing the direction of its concavity.

Answer 2

An inflection point, or point of inflection (or inflexion) can be defined in any of the following ways:

a point on a curve at which the tangent crosses the curve itself.
a point on a curve at which the curvature changes sign. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a vehicle along the curve, it is a point at which the steering-wheel is momentarily 'straight', being turned from left to right or vice versa.
a point on a curve at which the second derivative changes sign. This is very similar to the previous definition, since the sign of the curvature is always the same as the sign of the second derivative, but note that the curvature is not the same as the second derivative.
a point (x,y) on a function, f(x), at which the first derivative, f'(x), is at an extremum, i.e. a minimum or maximum. (This is not the same as saying that y is at an extremum, and in fact implies that y is not at an extremum).

Plot of y = x3, rotated, with tangent line at inflection point of (0,0).Note that since the first derivative is at an extremum, it follows that the second derivative, f''(x), is equal to zero, but the latter condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection. (An example of such a function is y = x4 − x).

It follows from the definition that the sign of f'(x) on either side of the point (x,y) must be the same. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.

Points of inflection can also be categorised according to whether f'(x) is zero or not zero.

if f'(x) is zero, the point is a stationary point of inflection, also known as a saddle-point
if f'(x) is not zero, the point is a non-stationary point of inflection

Plot of y = x4 - x with tangent line at non-inflection point of (0,0).An example of a saddle point is the point (0,0) on the graph y=x³. The tangent is the x-axis, which cuts the graph at this point.

A non-stationary point of inflection can be visualised if the graph y=x³ is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero.

Note that an inflection point is also called an ogee, although this term is sometimes applied to the entire curve which contains an inflection point.


See alsoIn mathematics, a critical point (or critical number) is a point on the domain of a function where the derivative is equal to zero or undefined. It is also called a stationary point.

For a function of several real variables, the condition of being a critical point is equivalent to all of its partial derivatives being zero; for a function on a manifold, it is equivalent to the exterior derivative being zero. For a map between spaces of arbitrary finite or infinite dimension, it means that the derivative is zero as a linear map.

If a critical point has a nonsingular Hessian matrix it is called nondegenerate, and the signs of the eigenvalues of the Hessian determine the function's local behavior. In the case of a real function of a real variable, the Hessian is simply the second derivative, and nonsingularity is equivalent to being nonzero. A nondegenerate critical point of a single-variable real function is a maximum if the second derivative is negative, and a minimum if it is positive. In general, the number of negative eigenvalues of a critical point is called its index, and a maximum occurs when all eigenvalues are negative (maximal index) and a minimum occurs when all eigenvalues are positive (index zero). Morse theory studies both finite and infinite dimensional manifolds using these ideas.

In one or several variables, the maxima and minima of a function (if they exist) can occur either at its critical points or at points on its boundary, or points where the function is not differentiable.

Critical points are also sometimes defined to be points where the derivative of a function is not of maximum rank, i.e. where it fails to be a submersion. The value of a function at a critical point, if defined, is called a critical value. Sard's theorem states that the set of critical values, in this sense of critical point, of a differentiable function has measure zero.

In the presence of a Riemannian metric or a symplectic form, to every smooth function is associated a vector field (the gradient or Hamiltonian vector field). These vector fields vanish exactly at the critical points of the original function, and thus the critical points are stationary, i.e. constant trajectories of the flow associated to the vector field.

Answer 3

Critical point is where combined shear stress and bending moments stress is maximum,
where as inflection point is where bendinh moment changes it sign

Answer 4

an inflection point is a point for a function f(x) such that

d/dx (df/dx) = - g(x),

where g(x) is a positive, or non negative, function. Basically, it means that at and after this certian point for a interval the rate at which the rate of change of f(x) is changing is negative. Aint never heard of critical point i reckons.

Answer 5

point of inflection is the point from where the direction in which slope changes is changed

critical point is the point where first derivative is:zero or undefined