Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f:R\to R$ be a function defined by $f(x)=\max \{x,x^3\}$. The set of all points where $f(x)$ is NOT differentiable is

Mastering the Max Function

Welcome, future engineers! Today, we are going to peel back the layers of a problem that often trips up students in the JEE Advanced exam. It involves the function $f(x)=\max \{x,x^3\}$.

At first glance, it looks simple, but it hides a beautiful geometric reality that tests your fundamental understanding of calculus. Let's embark on this journey together.

The Geometry of Competition

Imagine you are standing on the coordinate plane. You have two paths in front of you: the straight line $y=x$ and the cubic curve $y=x^3$.

The function $f(x)=\max \{x,x^3\}$ is like a traveler who is obsessed with altitude. At any given point $x$, this traveler chooses the path that is higher. If the line is above the curve, they walk on the line; if the curve rises above the line, they switch to the curve.

This is what we call the upper envelope of the two functions. To visualize this, sketch $y=x$ (the line passing through the origin at a $45^{\circ }$ angle) and $y=x^3$ (the curve that hugs the x-axis near the origin and then shoots up).

Notice how they weave around each other. They don't just meet once; they engage in a dance, crossing paths multiple times. Our goal is to find where this traveler experiences a 'jolt'—a sharp corner where the path switches abruptly.

The Intersection: Finding the Junctions

Before we can analyze the smoothness of the path, we must find where the traveler switches tracks. This happens exactly where the two paths meet. We set the two functions equal to each other:

Now, I want you to be careful here. Do not simply divide by $x$! If you do, you will lose the solution at $x=0$. Instead, bring everything to one side:

This gives us three critical junction points: $x=-1$, $x=0$, and $x=1$. These are the points where the 'max' function is forced to make a decision. At these points, the function switches its identity from one curve to the other.

The Calculus of Sharp Corners

In calculus, a function is differentiable at a point if it is smooth—meaning the slope is consistent as you approach from the left and the right. If the slope changes abruptly, you have a 'sharp corner' or a 'kink,' and the derivative does not exist.

Let's test our three junctions:

1. At $x=-1$: To the left, the line $y=x$ is higher. To the right, the cubic $y=x^3$ takes over.

The left-hand derivative ($LHD$) is:

The right-hand derivative ($RHD$) is:

At $x=-1$, $RHD=3(-1{)}^2=3$. Since $1 eq 3$, we have a sharp corner!

2. At $x=0$: To the left, $y=x^3$ is higher. To the right, $y=x$ is higher.

The $LHD$ is:

The $RHD$ is:

Since $0 eq 1$, we have another sharp corner!

3. At $x=1$: To the left, $y=x$ is higher. To the right, $y=x^3$ takes over again.

The $LHD$ is:

The $RHD$ is:

Since $1 eq 3$, we have our third sharp corner!

By systematically checking each junction, we have proven that at all three points—$x=-1$, $x=0$, and $x=1$—the function experiences a sudden change in slope.

Therefore, the set of all points where $f(x)$ is not differentiable is $\{-1,0,1\}$.

Remember, in JEE Advanced, the 'max' or 'min' function is a classic way to test if you truly understand the geometric meaning of a derivative. Don't just rely on algebra; visualize the curves, find the intersections, and check the slopes. You've got this!