critical point

Examples of critical point in the following topics:

• Maximum and Minimum Values

  • The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
  • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.
  • For example, if a bounded differentiable function $f$ defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem).
  • Its only critical point is at $(0,0)$, which is a local minimum with $f(0,0) = 0$.
  • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

• Concavity and the Second Derivative Test

  • The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
  • In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
  • The test states: if the function $f$ is twice differentiable at a critical point $x$ (i.e.
  • Telling whether a critical point is a maximum or a minimum has to do with the second derivative.
  • Calculate whether a function has a local maximum or minimum at a critical point using the second derivative test

• Optimization in Several Variables

  • To solve an optimization problem, formulate the function $f(x,y, \cdots )$ to be optimized and find all critical points first.
  • After finding out the function $f(x)$ to be optimized, local maxima or minima at critical points can be easily found.
  • (Of course, end points may have maximum/minimum values as well.)

• Maximum and Minimum Values

  • Maxima and minima are critical points on graphs and can be found by the first derivative and the second derivative.
  • The value of the function at this point is called maximum of the function.
  • Local extrema can be found by Fermat's theorem, which states that they must occur at critical points.
  • One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test.
  • Use the first and second derivative to find critical points (maxima and minima) on graphs of functions

• Applications of Minima and Maxima in Functions of Two Variables

  • In particular, we learned about the second derivative test, which is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum, using the value of the second derivative at the point.
  • Now we have to label the critical values using the second derivative test.
  • Plugging in all the different critical values we found to label them, we have:
  • At the remaining point we need higher-order tests to find out what exactly the function is doing.

• Optimization

  • After finding out the function $f(x)$ to be optimized, local maxima or minima at critical points can be easily found.
  • (Of course, end points may have maximum/minimum values as well.)

• Tangent Planes and Linear Approximations

  • The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.
  • The tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.
  • where $(x_0,y_0,z_0)$ is a point on the surface.
  • The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.

• Vectors in Three Dimensions

  • A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point $A$ with a terminal point $B$, and denoted by $\vec{AB}$.
  • For instance, in three dimensions, the points $A=(1,0,0)$ and $B=(0,1,0)$ in space determine the free vector $\vec{AB}$ pointing from the point $x=1$ on the $x$-axis to the point $y=1$ on the $y$-axis.
  • A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin $O = (0,0,0)$.
  • Thus the bound vector represented by $(1,0,0)$ is a vector of unit length pointing from the origin along the positive $x$-axis.
  • A vector in the 3D Cartesian space, showing the position of a point $A$ represented by a black arrow.

• The Derivative as a Function

  • If every point of a function has a derivative, there is a derivative function sending the point $a$ to the derivative of $f$ at $x = a$: $f'(a)$.
  • Let $f$ be a function that has a derivative at every point $a$ in the domain of $f$.
  • Because every point $a$ has a derivative, there is a function that sends the point $a$ to the derivative of $f$ at $a$.
  • Sometimes $f$ has a derivative at most, but not all, points of its domain.
  • Since $D(f)$ is a function, it can be evaluated at a point $a$.

• The Derivative and Tangent Line Problem

  • The tangent line $t$ (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • Informally, it is a line through a pair of infinitely close points on the curve.
  • To find the tangent line at the point $p = (a, f(a))$, consider another nearby point $q = (a + h, f(a + h))$ on the curve.
  • The line shows the tangent to the curve at the point represented by the dot.
  • Define a derivative as the slope of the tangent line to a point on a curve