absolute minimum
In mathematics, the term “absolute minimum” refers to the lowest value that a function or equation can attain over its entire domain
In mathematics, the term “absolute minimum” refers to the lowest value that a function or equation can attain over its entire domain. It is also known as the global minimum. To find the absolute minimum of a function, you typically need to determine the critical points and endpoints of the function, and then evaluate the function at these points to determine the absolute minimum.
Here is a step-by-step process to find the absolute minimum of a function:
1. Find the critical points: To find the critical points, you need to determine where the derivative of the function is equal to zero or undefined. Take the derivative of the function and set it equal to zero or find where it is undefined by solving any possible denominator equal to zero. Solve the resulting equation(s) to find the x-values of the critical points.
2. Find the endpoints: Determine the x-values at which the function is defined. These are the endpoints of the domain. If the function is defined on a closed interval, the endpoints will also need to be considered.
3. Evaluate the function at the critical points and endpoints: Substitute the x-values found in steps 1 and 2 into the original function to obtain the corresponding y-values.
4. Compare the y-values: Compare the y-values obtained from step 3 to determine the lowest y-value. The corresponding x-value of this y-value is the absolute minimum.
5. State the absolute minimum: State the x-value and the corresponding y-value as the absolute minimum of the function. For example, “The absolute minimum of the function f(x) is (x, y).”
It is important to note that finding the absolute minimum can be more complicated for more complex functions or when dealing with multivariable functions. In some cases, numerical methods or advanced techniques may be required to find the absolute minimum.
Overall, finding the absolute minimum involves locating the lowest value of a function over its entire domain and can be determined through a systematic process of finding critical points, endpoints, evaluating the function, and comparing the resulting values.
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