Given g(x)=f(x^2-x-10)+f(14+x-x^2) and f^''(x)>0 for all real x, except at finite no. of real va

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Given g(x)=f(x^2-x-10)+f(14+x-x^2) and f^''(x)>0 for all...

Given $g(x)=f\left(x^{2}-x-10\right)+f\left(14+x-x^{2}\right)$ and $f^{\prime \prime}(x)>0$ for all real $x$, except at finite no. of real values of $x$ for which $f^{\prime \prime}(x)=0$. Discuss the monotonicity of the function $g(x)$. \{ns. $(\infty,-3)$ decreasing, $\left(-3, \frac{1}{2}\right)$ increasing, $\left(\frac{1}{2}, 4\right)$ decreasing, $(4, \infty)$ increasing $\}$

Key Concepts

Function Composition in Derivative Analysis

Function composition involves applying one function to the result of another, and the derivative of the composite function is computed using the chain rule. When dealing with sums of composed functions, the derivative is the sum of the derivatives of each composed function. This technique is fundamental for analyzing the behavior of more complex functions formed by combining simpler functions.

Convexity and the Second Derivative Test

Convexity, often analyzed via the second derivative, is important in understanding the nature of a function's curvature. When a function’s second derivative is positive, the function is convex (or concave up), which can affect how the function reacts to changes in its inputs. The second derivative test is used to identify local minima or maxima and helps in determining the behavior of composite functions when combined with monotonicity analysis.

Critical Points and Interval Analysis

Identifying critical points, where the first derivative of a function is zero or undefined, is crucial in the analysis of the function's behavior. Once critical points are located, the overall monotonicity can be determined by testing the sign of the derivative on intervals defined by these points. This method allows one to systematically ascertain regions of increase and decrease within the function.

Monotonicity

Monotonicity refers to the behavior of a function in terms of increasing or decreasing. This analysis typically involves determining where the first derivative of the function is positive (indicating an increasing function) or negative (indicating a decreasing function). Establishing monotonicity is a central concept in understanding the overall behavior and trends of a function over various intervals.

Chain Rule

The chain rule is a fundamental principle in differentiation that is used when dealing with composite functions. It states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This principle is essential when differentiating functions where one function is applied to another.

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