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Absolute Extrema Finder

Find global min and max of quadratic functions on a closed interval

f(x) = ax^2 + bx + c on interval [L, R]

f(x)=x^2 +x + , interval [,]

Absolute Extrema Method

1. Find critical points: f(x)=0 inside [L,R]

2. Evaluate f at all critical points

3. Evaluate f at endpoints x=L and x=R

4. Largest = max, Smallest = min

Absolute extrema are the highest and lowest function values on a given interval. The Extreme Value Theorem guarantees their existence for continuous functions on closed intervals. Compare candidate values from critical points and endpoints.

⚠ Always check endpoints! The absolute extremum may occur at an endpoint rather than a critical point inside the interval.

What Are Absolute Extrema?

The absolute maximum is the largest y-value of the function on the interval. The absolute minimum is the smallest. To find them: evaluate the function at critical points (where f=0) and endpoints, then compare all values.

Critical Points

Where f(x)=0. For quadratics: f=2ax+b, critical point at x=-b/(2a). Candidates for extrema.

Endpoints

Evaluate f at both ends of the interval. The extremum can occur at boundaries, especially for monotonic functions.

Comparison

List all candidate y-values. The largest is absolute max. The smallest is absolute min. Both exist by EVT.

Extreme Value Theorem

Continuous function on [a,b] always attains global max and min. This theorem justifies the candidate comparison method.

Teaching Example: f(x)=x^2-4x+3 on [0,5]. f=2x-4=0 at x=2. f(0)=3, f(2)=-1, f(5)=8. Absolute min = -1 at x=2. Absolute max = 8 at x=5.

Applications

Optimization Engineering Economics Physics Data Science

Frequently Asked Questions

What is absolute extrema?▼

Global min and max on an interval. Find by checking critical points and endpoints, then comparing values.

Critical points?▼

Where f=0 or DNE. For quadratics: x=-b/(2a). Also check if the critical point lies inside the interval.

Why check endpoints?▼

The max/min might be at the boundary, not at a critical point. For monotonic functions, extrema are always at endpoints.

Extreme Value Theorem?▼

Continuous functions on closed intervals always have absolute max and min. Guarantees the search will succeed.

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