Question

Let $f(x) = a_0 + a_1|x| + a_2|x^2| + a_3|x^3|$, where $a_0, a_1, a_2, a_3$ are real constants. Then $f(x)$ is differentiable at $x = 0$

• A. whatever be a0,a1,a2,a3.
• B. for no values of a0,a1,a2,a3.
• C. Only if a1=0
• D. Only if a1=0, a3=0

Answer 1

The Correct Option is C

We are given the function: \[ f(x) = a_0 + a_1 |x| + a_2 |x^2| + a_3 |x^3| \] where \( a_0, a_1, a_2, a_3 \) are real constants. We need to determine the conditions under which \( f(x) \) is differentiable at \( x = 0 \). Step 1: Understanding the differentiability at \( x = 0 \) For \( f(x) \) to be differentiable at \( x = 0 \), the left-hand derivative and the right-hand derivative must exist and be equal at \( x = 0 \). In other words, the function must not have any sharp corners or discontinuities at \( x = 0 \). Step 2: Analyzing the behavior of the terms involving absolute values We break down the terms in \( f(x) \) and examine each one: 1. \( a_0 \) is a constant, so it does not affect differentiability. 2. \( a_1 |x| \): The function \( |x| \) is not differentiable at \( x = 0 \) because the left and right derivatives are not equal. 3. \( a_2 |x^2| \): Since \( |x^2| = x^2 \) for all \( x \), this term behaves like a smooth quadratic function, which is differentiable at \( x = 0 \). 4. \( a_3 |x^3| \): Since \( |x^3| = x^3 \) for all \( x \), this term behaves like a smooth cubic function, which is also differentiable at \( x = 0 \). Step 3: Differentiability condition at \( x = 0 \) The term \( a_1 |x| \) is the only term that could cause a problem with differentiability at \( x = 0 \). For \( f(x) \) to be differentiable at \( x = 0 \), we need the coefficient \( a_1 \) of \( |x| \) to be 0. This is because the function \( |x| \) has a non-differentiable corner at \( x = 0 \), and for differentiability, we must eliminate this term. Thus, \( f(x) \) will only be differentiable at \( x = 0 \) if \( a_1 = 0 \). Step 4: Conclusion Therefore, the correct condition for \( f(x) \) to be differentiable at \( x = 0 \) is: \[ \boxed{\text{Only if } a_1 = 0} \]