The graph of f lies entirely above the x axis and f^'(x)<0 for all x. (a) Give the critical poin

step 1 In this problem, we're given a function F of X, and we know a couple of the properties of this f

The graph of f lies entirely above the x axis and f^'(x)<0...

The graph of $f$ lies entirely above the $x$ axis and $f^{\prime}(x)<0$ for all $x$.
(a) Give the critical point(s) of the function, or explain how you know there are none.
(b) Say where the function increases and where it decreases.
$$y=f\left(x^{2}\right)$$

Key Concepts

Monotonic Functions

A monotonic function is one that is either entirely nonincreasing or nondecreasing over its domain. In this problem, the function f is given to be strictly decreasing since its derivative is always negative. This property is used to infer behavior and sign when composing f with another function, as well as determining where it might have extrema.

Critical Points

Critical points are the values in the domain of a function where its derivative is zero or undefined. In the context of the given composite function, identifying critical points involves setting the derivative equal to zero and using the definition of critical points in relation to the function’s behavior.

Chain Rule

The chain rule is a fundamental differentiation technique used to compute the derivative of a composite function. For a function defined as f(g(x)), the derivative is f'(g(x)) multiplied by g'(x). This rule is crucial in analyzing the derivative of the function f(x²), as it allows one to differentiate the outer function f and the inner function x² separately.

Increasing/Decreasing Behavior

The concepts of increasing and decreasing functions relate to the sign of the derivative. A function is increasing when its derivative is positive and decreasing when its derivative is negative. By analyzing the sign of the derivative (obtained via the chain rule) for f(x²), one can determine on which intervals the composite function increases or decreases.

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