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Minimum Point on A Closed Interval Calculator

Reviewed by Calculator Editorial Team

Finding the minimum point on a closed interval is a fundamental problem in calculus and optimization. This calculator helps you determine the minimum value of a function within a specified interval, which is useful in various mathematical and real-world applications.

What is a minimum point on a closed interval?

A minimum point on a closed interval [a, b] is the point x within the interval where the function f(x) attains its smallest value. This is different from a local minimum, which may not be the absolute minimum on the entire interval.

In calculus, the Extreme Value Theorem guarantees that a continuous function on a closed interval will have both a maximum and a minimum value. Finding these values is essential in optimization problems, physics, engineering, and economics.

How to find the minimum point

To find the minimum point on a closed interval, follow these steps:

  1. Identify the function f(x) and the interval [a, b].
  2. Find the critical points by solving f'(x) = 0 within the interval.
  3. Evaluate the function at the critical points and at the endpoints a and b.
  4. Compare these values to find the minimum.

Note: The function must be continuous on the closed interval and differentiable on the open interval (a, b) for this method to work.

Formula for minimum point

The minimum value of f(x) on [a, b] is the smallest value among:

  • f(a)
  • f(b)
  • f(c) where c is a critical point in (a, b)
min = min{f(a), f(b), f(c₁), f(c₂), ..., f(cₙ)} where cᵢ are critical points in (a, b)

Worked example

Let's find the minimum of f(x) = x³ - 3x² + 4 on the interval [0, 3].

  1. Find the derivative: f'(x) = 3x² - 6x.
  2. Find critical points: 3x² - 6x = 0 → x(3x - 6) = 0 → x = 0 or x = 2.
  3. Evaluate at critical points and endpoints:
    • f(0) = 0 - 0 + 4 = 4
    • f(2) = 8 - 12 + 4 = 0
    • f(3) = 27 - 27 + 4 = 4
  4. The minimum value is 0 at x = 2.

FAQ

What if the function has no critical points in the interval?
If there are no critical points, the minimum must occur at one of the endpoints. Simply compare f(a) and f(b).
Can I use this calculator for non-continuous functions?
No, this calculator assumes the function is continuous on the closed interval. For discontinuous functions, you'll need a different approach.
What if the function has multiple minima?
The calculator will find the global minimum on the interval. If you need all local minima, you would need additional analysis.