Math Problem Statement

Find the range of a such that g(x) reaches its minimum, and express this minimum in terms of a. The function g(x) is defined as g(x) = ∫ from -2 to 2 |x - t| * (t^2 - a^2) dt.

Solution

Let's break down the problem step-by-step:

Given: g(x)=22xt(t2a2)dt,g(x) = \int_{-2}^{2} |x - t| (t^2 - a^2) \, dt, we need to find the range of aa such that g(x)g(x) has its minimum value, and also express this minimum value in terms of aa.

Step-by-step Analysis:

  1. Understanding g(x)g(x): g(x)g(x) involves an integral from 2-2 to 22 of the product of two functions:

    • xt|x - t|, which represents the absolute difference between xx and tt.
    • (t2a2)(t^2 - a^2), a quadratic function that depends on the parameter aa.
  2. Behavior of the Integral:

    • The function xt|x - t| changes its behavior depending on whether t<xt < x or txt \geq x.
    • The function (t2a2)(t^2 - a^2) changes sign depending on whether t<a|t| < |a| or ta|t| \geq |a|.
  3. Split the Integral: Since xt|x - t| can be split into different cases:

    • When t<xt < x: xt=xt|x - t| = x - t.
    • When txt \geq x: xt=tx|x - t| = t - x.

    Depending on xx, split the integral: g(x)=2x(xt)(t2a2)dt+x2(tx)(t2a2)dt.g(x) = \int_{-2}^{x} (x - t)(t^2 - a^2) \, dt + \int_{x}^{2} (t - x)(t^2 - a^2) \, dt.

  4. Finding the Minimum Value: To find the minimum value of g(x)g(x), consider the integrand's symmetry and behavior as a function of aa. The minimum value will depend on the sign of (t2a2)(t^2 - a^2) and how it interacts with xt|x - t|.

  5. Conditions on aa:

    • If a2<4a^2 < 4, then t2a2t^2 - a^2 will be negative for some parts of the interval [2,2][-2, 2].
    • If a24a^2 \geq 4, then t2a2t^2 - a^2 is non-positive for the entire range of integration.

Conclusion:

  1. Finding the Range for aa:
    • The range of aa for which g(x)g(x) achieves its minimum is when a24a^2 \geq 4, or equivalently a2|a| \geq 2.
  2. Expressing Minimum Value:
    • When a2|a| \geq 2, g(x)g(x) reaches a minimum at some specific values of xx (likely by symmetry, possibly at x=0x = 0).

Final Answer:

  1. The range of aa such that g(x)g(x) has its minimum value: a2|a| \geq 2.
  2. Express g(x)g(x)'s minimum value in terms of aa.

Let me know if you want to go deeper into the integration process or have any other questions!

Related Questions:

  1. How do you handle integrals with absolute value functions?
  2. What is the geometric interpretation of g(x)g(x) in this context?
  3. How does the choice of aa affect the sign of the integrand?
  4. What is the role of symmetry in finding the minimum value of this integral?
  5. How can you simplify the integral if x=0x = 0?

Tip:

When dealing with integrals that include absolute values, always consider breaking the integral into different intervals where the absolute value expression can be simplified.

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Math Problem Analysis

Mathematical Concepts

Calculus
Definite Integrals
Optimization
Absolute Value
Quadratic Functions

Formulas

g(x) = ∫ from -2 to 2 |x - t| * (t^2 - a^2) dt

Theorems

Fundamental Theorem of Calculus
Properties of Absolute Value
Optimization with respect to parameters

Suitable Grade Level

University Level