f(x)={ 2-|x^2+5 x+6|, x ≠-2 a^2+1 , x=-2 ., then the range of a so that f(x) h

step 1 So in this question we have f of x f of x is equal to is equal to two minus two minus more x plu

F(x)={ 2-|x^2+5 x+6|, x ≠-2 a^2+1 ,...

$f(x)=\left\{\begin{array}{cc}2-\left|x^{2}+5 x+6\right|, & x \neq-2 \\ a^{2}+1 & , x=-2\end{array}\right.$, then the
range of a so that $f(x)$ has maxima at $x=-2$ is $($ A) $|a| \geq 1$
(B) $\mid \mathrm{a}<1$
(C) $\mathrm{a}>1$
(D) $\mathrm{a}<1$

Key Concepts

Piecewise Functions

Piecewise functions are defined by different formulas over different parts of their domain. Analyzing such functions requires evaluating each piece separately and understanding how their definitions interact at the boundaries of their intervals to determine overall properties like continuity, differentiability, and extremum points.

Maximum Values (Extremum Conditions)

Determining a maximum (or minimum) of a function involves checking that the function’s value at a specific point is greater (or lesser) than at nearby points and over the entire domain. For functions defined piecewise, this often requires comparing the value at the point of interest with limits or values from other pieces to ensure the extremum condition is met.

Absolute Value Functions

Absolute value functions create piecewise definitions by considering the positive and negative cases of the expression inside the absolute value. Understanding how an absolute value affects an expression, such as making a quadratic expression non-negative, is key when evaluating and analyzing the behavior of a function.

Quadratic Functions

Quadratic functions are polynomials of degree two that can often be factored or completed in square form. Their standard forms help identify roots, vertex, and axis of symmetry, which are crucial when determining the behavior of the function, especially in the context of maximum or minimum values.

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