Let f(x)=max(x, x^2, x^3) in -2 ≤x ≤2. Then (a) f(x) is continuous in -2 ≤x ≤2 (b) f(x) is not d

step 1 about the number 297 question says f x equal to we have to assume fx equal to maximum of x x squ

Let f(x)=max(x, x^2, x^3) in -2 ≤x ≤2. Then (a) f(x) is...

Let $\mathrm{f}(x)=\max \left(\mathrm{x}, \mathrm{x}^{2}, \mathrm{x}^{3}\right)$ in $-2 \leq \mathrm{x} \leq 2$. Then (a) $\mathrm{f}(\mathrm{x})$ is continuous in $-2 \leq \mathrm{x} \leq 2$ (b) $\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=1$ (c) $\mathrm{f}(-1)+\mathrm{f}\left(\frac{3}{2}\right)=\frac{35}{8}$ (d) $f^{\prime}(-1) f^{\prime}\left(\frac{3}{2}\right)=\frac{-35}{4}$

Let $\mathrm{f}(x)=\max \left(\mathrm{x}, \mathrm{x}^{2}, \mathrm{x}^{3}\right)$ in $-2 \leq \mathrm{x} \leq 2$. Then
(a) $\mathrm{f}(\mathrm{x})$ is continuous in $-2 \leq \mathrm{x} \leq 2$
(b) $\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=1$
(c) $\mathrm{f}(-1)+\mathrm{f}\left(\frac{3}{2}\right)=\frac{35}{8}$
(d) $f^{\prime}(-1) f^{\prime}\left(\frac{3}{2}\right)=\frac{-35}{4}$

Key Concepts

Maximum Function

A maximum function takes the highest value from a set of functions at each point in the domain. When designing or analyzing such functions, it is important to compare the constituent functions across intervals to determine which one is dominant, resulting in a piecewise form of the overall function.

Continuity

Continuity refers to a function's property of having no breaks, jumps, or holes in its graph. In the context of maximum functions, since the constituent functions are continuous, the overall function is typically continuous, though care must be taken at the junctions where the maximum switches from one function to another.

Differentiability

Differentiability involves whether a function has a well-defined derivative at a point. For piecewise functions derived from maximum operations, differentiability must be examined particularly at the boundary points where the active function changes, as these are common sites for potential 'sharp corners' where the derivative may not exist.

Piecewise Function Definition

When a function is defined as the maximum of several functions, it naturally forms a piecewise function where different expressions define the function in different intervals. Analyzing these intervals separately is crucial to understanding the overall behavior, including continuity, differentiability, and the calculation of derivatives.

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