Question

Let \(f:[a, b] \rightarrow[1, \infty)\) be a continuous function and let \(g: \mathbb{R} \rightarrow \mathbf{R}\) be defincd as $$ g(x)= \begin{cases}0 & \text { if } xb\end{cases} $$ Then ( A) \(\mathrm{g}(\mathrm{x})\) is continuous but not differentiable at \(\mathrm{a}\) (B) \(\mathrm{g}(\mathrm{x})\) is differentiable on \(\mathbb{R}\) (C) \(\mathrm{g}(\mathrm{x})\) is continuous but not differentiable at \(\mathrm{b}\) (D) \(\mathrm{g}(\mathrm{x})\) is continuous and differentiable at either a or but not both

Short answer

Option (B) is true; g(x) is differentiable on \(\mathbb{R}\) which includes being continuous and differentiable at both a and b, satisfying (D) as well, implying (A) is incorrect since g(x) is differentiable at a, and (C) is incorrect as g(x) is differentiable at b.

Step-by-step solution

- Analyze the Definition of g(x)

Start by examining the definition of the function g. It is defined piecewise with three distinct expressions, one for each interval: x < a, a ≤ x ≤ b, and x > b.

- Continuity at x = a

Evaluate the limit of g(x) as x approaches a from the right. Since f is continuous on [a, b], by the Fundamental Theorem of Calculus, g(x) will also be continuous. Because g(x) is defined to be 0 for x < a, it matches with g(a) = 0 when approaching a from the left, confirming it is continuous at x = a.

- Differentiability at x = a

Investigate if g(x) has a derivative at x = a. By Fundamental Theorem of Calculus, g(x) is differentiable on (a, b), but for x = a, g(x) is not differentiable since the left-hand limit of the derivative does not exist (the function g(x) is constant for x < a).

- Continuity at x = b

Next, examine the continuity of g(x) at x = b. Similar to the argument at x = a, as f is continuous on [a, b], g(x) will be continuous at x = b too.

- Differentiability at x = b

Finally, assess the differentiability at x = b. g(x) is differentiable on (a, b), and since the value of g(x) becomes constant for x > b, the derivative of g(x) at x = b will exist and be equal to 0.

- Conclude g(x)'s Properties

After analyzing the function's continuity and differentiability on the intervals and at the points a and b, we can conclude that g(x) is continuous on \(\mathbb{R}\) and differentiable on \(\mathbb{R}\) except at x = a. Therefore, Option (C) is incorrect, Option (A) and (D) could be correct based on previous steps, and Option (B) is true.

Key concepts

Continuity and Differentiability

Understanding the concepts of continuity and differentiability is fundamental to studying calculus.

Continuity at a point means that there is no abrupt change in the value of the function at that point. To test for continuity at a point, you should look for three things: First, the function must be defined at the point. Second, the limit of the function as it approaches the point from both directions must exist. Third, these two values should be equal (the value of the function equals the limit).

Differentiability, on the other hand, refers to the existence of a derivative at a point. If a function is differentiable at a point, then it also continuous at that point, but the converse may not be true. This aspect is crucial in analyzing the differentiability of piecewise functions, such as the one given in the exercise. The derivative measures the rate at which the function's value is changing at an instant, or in simpler terms, it's the slope of the tangent line to the function's graph at that point.

When combining these two concepts, we find a crucial connection. A function can be continuous at a point but not differentiable there. This happens in situations where there's a sharp corner or a cusp in the graph of the function, or if the function has a vertical tangent at that point.

In our example, at point a, the function g(x) is continuous but not differentiable due to the abrupt change in behavior—from a constant value to an integral expression—which eliminates the possibility of a smooth tangent being drawn at that point.

Piecewise Functions

A piecewise function is defined by different expressions for different intervals of its domain. It is like a mathematical quilt, stitched together using various functional pieces to form a unified whole.

One crucial aspect of dealing with piecewise functions is ensuring that they are well-defined at the boundaries of these pieces. This involves checking for continuity, which often means verifying that the function approaches the same value from both sides of the boundary.

In the context of the JEE Advanced mathematics problem given, the function g(x) is defined in a piecewise manner based on the value of x relative to a and b. It behaves differently on the intervals x < a, a ≤ x ≤ b, and x > b, which requires careful examination when determining its continuity and differentiability.

The fundamental theorem of calculus plays a pivotal role in understanding these changing behaviors, particularly when x is between a and b, as it relates the integral of a function to its antiderivative, allowing us to study its properties across the different intervals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between differential and integral calculus. This theorem has two parts. The first part states that if f is continuous on [a, b], then the function g, defined as the integral of f from a to x, is continuous on [a, b] and differentiable on (a, b), and its derivative is f(x).

The second part tells us that the integral of a function f from a to b is the antiderivative of f evaluated at b minus the antiderivative evaluated at a.

Applying this theorem to our piecewise function g(x) helps us determine its continuity and differentiability. The function g is defined as the integral of f from a to x for a ≤ x ≤ b, so by the first part of the Fundamental Theorem of Calculus, g is continuous and differentiable in this interval, and its derivative is simply f(x). As for the interval x > b, g(x) remains constant, and hence, its derivative is 0, which guarantees differentiability at x = b.

This theorem is the key to unlocking the behavior of the function across different intervals and is central to solving the problem at hand.