Problem 40 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}\) be differentiable at \(\mathrm{c} \in \mathbf{R}\) and \(\mathrm{f}(\mathrm{c})=0\). If \(g(x)=|f(x)|\), then at \(x=c\), gis: \(\quad[\) (a) not differentiable if \(f^{\prime}(c)=0\) (b) differentiable if \(\mathrm{f}^{\prime \prime}(\mathrm{c}) \neq 0\) (c) differentiable if \(\mathrm{f}^{\prime}(\mathrm{c})=0\) (d) not differentiable

Step-by-Step Solution

Answer

Option (c) is correct: g is differentiable if \(f'(c) = 0\).

Step 1: Understanding the Function g(x)

The function given is \(g(x) = |f(x)|\). This function involves absolute value, which can cause issues at points where \(f(x)\) transitions from positive to negative or vice versa.

Step 2: Analyze g(x) at x=c

We know \(f(c) = 0\), so \(g(c) = |f(c)| = 0\). The point \(c\) is crucial because it is a point where the behavior of \(f(x)\) around \(x = c\) affects the differentiability of \(g(x)\).

Step 3: Consider Differentiability of g(x)

The differentiability of \(g(x)\) at \(x = c\) depends on the derivative of \(f(x)\) at that point. When we have \(|f(x)|\) and \(f(c) = 0\), \(g(x)\) is differentiable at \(x = c\) if the left-hand and right-hand derivatives of \(f(x)\) are equal at \(c\).

Step 4: Apply Differentiability Conditions

For \(g(x) = |f(x)|\) to be differentiable at \(x = c\), \(f(x)\) must be zero at \(x = c\) and \(\frac{d}{dx} f(x)\) must also be zero (i.e., \(f'(c) = 0\)). If \(f'(c) eq 0\), \(g(x)\) has a cusp at \(c\), making \(g(x)\) non-differentiable.

Step 5: Examine Given Options

Given the options, (c) states \(g(x)\) is differentiable if \(f'(c) = 0\). This is true because if \(f'(c) = 0\), the function \(g(x)\) can be smoothly tangent at \(x = c\). Thus, (c) is correct.

Key Concepts

Absolute Value Function

An absolute value function, like \(g(x) = |f(x)|\), takes any input and outputs its non-negative magnitude. This function has unique properties, especially when dealing with differentiability.
Unlike standard linear equations, the absolute value function can create sharp corners, or cusps, at points where the input function \(f(x)\) switches sign.

• This sign change occurs when \(f(x)\) transitions through zero, such as at \(f(c) = 0\) in our example.
• At such points, the absolute value function can either be smooth or present a sudden change in direction, affecting its differentiability.

Understanding the behavior of absolute value functions, particularly at these key points, is crucial for analyzing complex functions that involve them.

Derivative

The concept of a derivative is central to understanding differentiability. It measures how a function changes as its input changes, often conceived as the slope of the function at any point.
In our exercise, we're interested in the behavior of \(g(x) = |f(x)|\) around \(x = c\) where \(f(c) = 0\).

• If \(f'(c) = 0\), it suggests that the function \(f(x)\) is flat at \(x = c\), potentially allowing \(g(x)\) to be differentiable there.
• If \(f'(c) \eq 0\), the function may not smoothly transition to the absolute value, resulting in a cusp rather than a smooth curve.

Derivatives help in examining these subtle changes, and their values are instrumental in determining the differentiability of functions involving absolute values.

Cusp

A cusp is a point on a graph where the curve stops behaving smoothly and instead forms a sharp corner. It is a significant feature when we talk about the differentiability of functions like \(g(x) = |f(x)|\).
At a cusp, a function doesn't have a clear tangent line. Instead, both the left-hand and right-hand derivatives exist but are not equal.

• This is usually observed when \(f(x)\) changes sign at a point such as \(x = c\), making \(|f(x)|\) non-differentiable at that cusp.
• If \(f'(c) \eq 0\), it's indicated that the cusp occurs, adding complexity to finding derivatives at these tricky points.

Thus, recognizing cusps is essential for understanding when and why certain functions are non-differentiable.

Differentiable Function

A function is considered differentiable at a point if it has a defined, continuous derivative at that point. This means there's a tangent line that fits the curve smoothly.
For the function \(g(x) = |f(x)|\), determining differentiability involves analyzing the derivatives of \(f(x)\) itself.

• If \(f'(c) = 0\), \(g(x)\) is differentiable at \(x = c\) because the point \(f(c) = 0\) allows the curves to meet without forming a cusp.
• In contrast, if \(f'(c) \eq 0\), the function \(g(x)\) is likely to have a cusp, rendering it non-differentiable at \(x = c\).

Understanding the conditions under which a function remains differentiable aids in distinguishing between smooth functions and those that appear jagged.