If x ∈[0,2] and g(x)=f(x)+f(2-x). Also, f^''(x)<0 then g(x) (A) increases in [0,2] (B) decreases

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If x ∈[0,2] and g(x)=f(x)+f(2-x). Also, f^''(x)<0 then g(x)...

If $x \in[0,2]$ and $g(x)=f(x)+f(2-x)$. Also, $f^{\prime \prime}(x)<0$
then $g(x)$
(A) increases in $[0,2]$
(B) decreases in $[0,2]$(C) decreases in $[0,1)$ and increases in $(1,2]$
(D) increases in $[0,1)$ and decreases in $(1,2]$

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Key Concepts

Symmetry in Composite Functions

The function g(x) is defined as the sum of f(x) and f(2-x), which creates a reflective symmetry about the point x=1. This type of symmetry is important because it often leads to a balanced or mirrored behavior in the function's properties, such as having a maximum or minimum at the midpoint of the interval.

Monotonicity Analysis Using Derivatives

By differentiating g(x) to obtain g'(x)= f'(x) - f'(2-x), one can study the function's monotonicity. Given that f' is a decreasing function, it can be deduced that for x less than 1, f'(x) is relatively larger than f'(2-x), resulting in a positive derivative (increasing g), and for x greater than 1, the derivative turns negative (decreasing g). This derivative test is a key tool in analyzing where a function increases or decreases.

Concavity and Its Implications

The fact that f''(x) is negative indicates that f is concave downward on its domain. This means that the first derivative f' is decreasing. Such a property is crucial when analyzing the behavior of functions composed using f, as the reduction in the derivative provides information about the rate at which f increases or decreases over an interval.

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