If g(x)=f(x)+f(1-x) and f^''(x)<0 for 0 ≤x ≤1, then (A) g(x) increases in (-∞, (1)/(2)) (B) g(x)

step 1 F double dash x is less than 0. In interval 0 to 1, this implies f -dash -x is decreasing functi

If g(x)=f(x)+f(1-x) and f^''(x)<0 for 0 ≤x ≤1, then (A) g(x)...

If $g(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x)<0$ for $0 \leq x \leq 1$, then (A) $g(x)$ increases in $\left(-\infty, \frac{1}{2}\right)$ (B) $g(x)$ increases in $\left(0, \frac{1}{2}\right)$ (C) $g(x)$ decreases in $\left(\frac{1}{2}, 1\right)$ (D) $g(x)$ decreases in $\left(\frac{1}{2}, \infty\right)$

If $g(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x)<0$ for $0 \leq x \leq 1$, then
(A) $g(x)$ increases in $\left(-\infty, \frac{1}{2}\right)$
(B) $g(x)$ increases in $\left(0, \frac{1}{2}\right)$
(C) $g(x)$ decreases in $\left(\frac{1}{2}, 1\right)$
(D) $g(x)$ decreases in $\left(\frac{1}{2}, \infty\right)$

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Key Concepts

First Derivative and Monotonicity

The first derivative of a function indicates its rate of change. If the derivative is positive, the function is increasing; if it is negative, the function is decreasing. In problems like the one given, analyzing the sign of the derivative of a composite function can help determine the intervals on which the function increases or decreases.

Concavity

Concavity refers to the curvature of a function. When a function is concave down (as indicated by its second derivative being negative), it means that the slope or first derivative of the function is decreasing. This property is crucial because it allows us to conclude that the change in the function’s rate of increase or decrease is in a specific direction, which in turn influences the monotonic behavior of related functions.

Function Symmetry and Composition

The function defined as g(x) = f(x) + f(1-x) exhibits a form of symmetry because the roles of x and 1 - x are interchangeable. This symmetry can lead to specific properties, such as the derivative being zero at the midpoint of the interval, and helps in splitting the analysis into intervals based on this symmetric behavior.

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