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Critical points


Instructions:

Write the coefficients of the quadratic equation in boxes \(a\), \(b\), \(c\), \(d\), and \(e\) to change the polynomial. Check the other box to graph the function derivative.


Explanation

To find the maximum and minimum points and inflection points of a function \(f\left(x\right)\), follow these steps:

  1. First Derivative for maximum and minimum of a function
    1. Find the first derivative \(f^{\prime}(x)\) of the function \(f(x)\).
    2. Set the derivative to zero and solve for \( x : f^{\prime}\left( x \right) = 0\). This will give you the critical points.
    3. Verify the critical points in the original function to determine if they are maxima, minima, or inflection points. Also, evaluate the values of \( f\left(x\right) \) at the boundaries of the domain if necessary.
  2. Second Derivative and Inflection Points
    1. Find the second derivative \( f^{\prime\prime}\left(x\right) \) of the function \(f\left(x\right)\).
    2. Set the second derivative to zero and solve for \(x: f^{\prime\prime}(x)=0\). This will give you possible inflection points.
    3. Verify the points where \(f^{\prime\prime}\left(x\right) = 0\) to confirm if they are inflection points by checking for a change in concavity. A change in the sign of \(f^{\prime\prime}\left(x\right)\) across the point indicates an inflection point.
  3. Classification of Critical Points
  4. To classify the critical points \(x = c\) found:

    1. Second Derivative Test
      • If \(f^{\prime\prime}\left(c\right) > 0\), \(x=c\) is a local minimum.
      • If \(f^{\prime\prime}\left(c\right) < 0\), \(x=c\) is a local maximum.
      • If \(f^{\prime\prime}(c)=0\), the test is inconclusive, and other methods like the first derivative test or analyzing the sign of \(f^{\prime\prime}\left(x\right)\) around the critical point can be used.
    2. First Derivative Test
      • If \(f^{\prime}\left(x\right)\) changes from positive to negative at \(x=c\), then \(x=c\) is a local maximum.
      • If \(f^{\prime}\left(x\right)\) changes from negative to positive at \(x=c\), then \(x=c\) is a local minimum.

See also

Box optimization

Differential Calculus

Path optimization