Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)=\sum_{k=0}^{n} a_{k}|x-1|^{k}\), where \(a_{i} \in R\) then (A) \(f(x)\) is continuous at \(x=1\) for all \(a_{k} \in R\) (B) \(f(x)\) is differentiable at \(x=1\) for all \(a_{k} \in R\) (C) \(f(x)\) is differentiable at \(x=1\), provided \(a_{2 k+1}=0\) (D) \(f(x)\) is continuous at \(x=1\), provided \(a_{2 k}=0\)

Step-by-Step Solution

Answer

(A) and (C) are true.

Step 1: Understanding the Function

The given function is defined as a finite sum, \(f(x) = \sum_{k=0}^{n} a_{k}|x-1|^{k}\). The function uses absolute values, which can affect continuity and differentiability at certain points, especially where the argument of the absolute value is zero.

Step 2: Checking Continuity at \(x=1\)

For continuity at \(x=1\), the left-hand limit, right-hand limit, and the function value at \(x=1\) must be equal. At \(x=1\), \(|x-1| = 0\), making \(f(x) = a_0\) for any \(a_{k}\). Since \(a_0\) is a real number, \(f(x)\) is continuous at \(x=1\) for any coefficient \(a_{k} \in \mathbb{R}\). Hence, option (A) is true.

Step 3: Differentiability at \(x=1\)

To be differentiable at \(x=1\), the function must be smooth and not have a corner or cusp at this point. Due to the absolute value, \(|x-1|^{k}\), terms can introduce such non-smooth points. However, if the terms with odd powers are zero \((a_{2k+1}=0)\), the function at \(x=1\) becomes even-powered, making it symmetrical and smooth. Thus, \(f(x)\) will be differentiable at \(x=1\) if \(a_{2k+1}=0\). Hence, option (C) is true.

Step 4: Conclusion

Both conditions for continuity in option (A) and differentiability in option (C) involve different sets of coefficients. Only option (A) depends on all \(a_k\) being real numbers without additional conditions, while (C) requires elimination of terms to maintain differentiability.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It is denoted by \(|x|\), which represents the distance of a number from zero on the number line, without considering direction. More formally, for any real number \(x\), the absolute value function is defined as follows:
\[ |x| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases} \]
The absolute value function is always non-negative.
It has a characteristic "V" shape when graphed, containing a sharp turning point at the origin (\(x = 0\)).
This turning point often causes issues with differentiability, as seen in the step-function behavior near such points. In contrast, absolute value does not hinder continuity as it naturally bridges over the origin part of the graph. Understanding how absolute values affect mathematical properties such as continuity and differentiability is crucial when analyzing functions, especially those that incorporate terms with absolute values in higher powers.

Continuity at a Point

Continuity of a function at a point, say \(x = a\), is an essential concept in calculus and mathematical analysis. A function \(f(x)\) is continuous at \(x = a\) if the following three conditions are satisfied:

• The function \(f(x)\) is defined at \(x = a\).
• The limit of \(f(x)\) as \(x\) approaches \(a\) from both directions exists.
• The left-hand and right-hand limits of the function at \(x = a\) are equal to the function’s value at that point (i.e., \(\lim_{x \to a^{-}}f(x) = \lim_{x \to a^{+}}f(x) = f(a)\)).

For the given function \(f(x) = \sum_{k=0}^{n} a_{k}|x-1|^{k}\), checking continuity at \(x = 1\) involves evaluating whether the above conditions are met.
When \(x = 1\), the term \(|x-1|\) becomes zero, simplifying \(f(x)\) to the constant value \(a_0\).
As long as \(a_0\) is a real number, \(f(x)\) is continuous at \(x = 1\).
This demonstrates that for all real coefficients \(a_k\), the function is continuous at the specified point without any additional conditions required.

Differentiability at a Point

Differentiability at a point is a more stringent condition than continuity.
A function \(f(x)\) is said to be differentiable at \(x = a\) if it has a well-defined, single-valued derivative at that point.
This means \(f(x)\) must have a smooth, non-vertical tangent line at \(x = a\).
In terms of formal definition, differentiability implies continuity, but the reverse is not always true.
For the function \(f(x) = \sum_{k=0}^{n} a_{k}|x-1|^{k}\), differentiability at \(x=1\) can be tricky because the absolute value function introduces potential non-smooth points.
These are often caused by the abrupt direction changes inherent in the function.
For the function to be differentiable at \(x = 1\), it is necessary that the terms responsible for introducing sharpness—those with odd powers—are neutralized.
This is achieved by ensuring that \(a_{2k+1} = 0\), which removes asymmetric non-smoothness, leading the remaining terms with even powers to dictate the function's behavior.
Thus, only terms that form a symmetrical, smooth curve around \(x=1\) remain, enabling differentiability at this point.