The function defined by f(x)=max{x^2,(x-1)^2. 2 x(1-x)}, 0 ≤x ≤1 (a) is differentiable for all x

step 1 In this problem we have been given let f be a function such that f minus 1 to 1 by 2 to pi by 2

The function defined by f(x)=max{x^2,(x-1)^2. 2 x(1-x)}, 0...

The function defined by $f(x)=\max \left\{x^{2},(x-1)^{2}\right.$ $2 x(1-x)\}, 0 \leq x \leq 1$ (a) is differentiable for all $x$ (b) is differentiable for all $x$ except at one point (c) is differentiable for all $x$ except at two points (d) is not differentiable at more than two points

The function defined by $f(x)=\max \left\{x^{2},(x-1)^{2}\right.$ $2 x(1-x)\}, 0 \leq x \leq 1$
(a) is differentiable for all $x$
(b) is differentiable for all $x$ except at one point
(c) is differentiable for all $x$ except at two points
(d) is not differentiable at more than two points

Key Concepts

Differentiability

This concept deals with whether a function has a defined tangent (slope) at every point in its domain. For a function to be differentiable at a point, it must be locally linearizable around that point, meaning that the limit defining the derivative must exist and be the same from both sides. The study of differentiability is essential when analyzing functions, especially where different formulas may govern behavior in various regions.

Maximum of Functions

Taking the maximum of several functions creates a new function that, at each point, equals the largest of the involved function values. This process often results in a piecewise definition where different underlying functions dictate the value on different subintervals. While each individual function might be smooth, the maximum function can suffer from non-differentiability at points where the leading function changes.

Piecewise Defined Functions

These are functions defined by different expressions over distinct parts of their domain. In many cases, even if every component function is continuous or differentiable, abrupt transitions at the junctions can result in discontinuities in the derivative. Therefore, understanding piecewise definitions is crucial in evaluating the overall smoothness and differentiability of such functions.

Intersection Points of Component Functions

The intersection points of the functions that are being compared play a key role in determining where the overall maximum function might switch from one rule to another. At these points, even though the values of the functions may agree, their derivatives might not match, potentially leading to corners or cusps. Such points are often the root cause of non-differentiability in maximum functions.

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