Question

Let \(a, b \in \mathbb{R}\) and \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=a \cos \left(\left|x^{3}-x\right|\right)+b|x| \sin \left(\left|x^{3}+x\right|\right) .\) Then \(f\) is (A) differentiable at \(x=0\) if \(a=0\) and \(b=1\) (B) differentiable at \(x=1\) if \(a=1\) and \(b=0\) (C) NOT differentiable at \(x=0\) if \(a=1\) and \(b=0\) (D) NOT differentiable at \(x=1\) if \(a=1\) and \(b=1\)

Short answer

Option (A) is correct: f is differentiable at x=0 if a=0 and b=1 because the cosine term becomes constant and the sine term retains x, resulting in a differentiable function. Differentiability of the other conditions can be determined using the derivative evaluated at specific points and analyzing continuity.

Step-by-step solution

Analyze Function at x=0

To determine differentiability at a point, we need to check whether the derivative exists at that point. For the function to be differentiable at x=0, the limit of the derivative as x approaches 0 must exist. Consider the function f(x) given by the equation and find f'(x) using the chain rule and the product rule as necessary.

Compute the Derivative

Differentiating f(x) yields two main components from the sum: one involving the derivative of the cosine term, and one involving the derivative of the sine term. Each involves the chain rule due to the absolute value and the product rule due to the factors a and b.

Simplify the Derivative

Simplify the expression obtained for f'(x), keeping in mind that the derivative of absolute value function |x| is not well-defined at x=0. The derivative of sin and cos are \(\sin(x)' = \cos(x)\) and \(\cos(x)' = -\sin(x)\), respectively. Additionally, differentiate the absolute value function using piecewise derivatives, based on the interval x is in.

Evaluate the Limit of the Derivative at x=0

We have to check the left-hand limit and the right-hand limit of the derivative as x approaches 0. If they are equal, f'(0) exists and f(x) is differentiable at x=0. If they diverge, the function is not differentiable at that point.

Examine Differentiability at x=1

Similar procedure needs to be applied at x=1. We need to check the derivative's behavior as x approaches 1 from both sides.

Consider Given Conditions

Using the values of a and b provided in the options (A), (B), (C), and (D), determine whether the derivative exists and is continuous at x=0 and x=1 respectively.

Key concepts

Differentiation Rules

Understanding differentiation rules is essential when dealing with complex functions. At its core, differentiation is about finding the rate at which a function changes at any given point. It's a fundamental tool in calculus used to determine the slope of a tangent to a function's graph and allows us to compute velocity, acceleration, and many other rates of change in various fields.

There are several basic differentiation rules that one must grasp:

• Constant Rule: The derivative of a constant is zero.
• Power Rule: For a function of the form x^n, the derivative is nx^(n-1).
• Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
• Product Rule: It allows the differentiation of a product of two functions, which we will elaborate on in a separate section.
• Quotient Rule: It is useful when dividing two functions.
• Chain Rule: Essential for differentiating composite functions, also to be discussed in detail shortly.

Applying these rules becomes second nature with practice, which is critical when determining the differentiability of a function at a certain point, such as in the given exercise.

Chain Rule

The chain rule is a powerful differentiation tool used when dealing with composite functions, which are functions composed of functions within functions. When a function f(x) can be expressed as g(h(x)), the derivative of f(x) with respect to x is the derivative of g with respect to h(x) times the derivative of h with respect to x. Mathematically, we write this as (f(g(x)))' = f'(g(x)) * g'(x).

For example, if you need to differentiate f(x) = cos(|x^3 - x|), you treat |x^3 - x| as an inner function (say h(x)) and cos(u) as an outer function (say g(u)). The chain rule is then applied to compute the derivative, which is crucial in the exercise we are considering.

Product Rule

The product rule is another essential differentiation principle used when a function is the product of two or more functions. The rule states that the derivative of a product of functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. In other words, if u(x) and v(x) are two functions, their product u(x) * v(x) has a derivative u'(x)v(x) + u(x)v'(x).

This rule is particularly useful when dealing with the function f(x) = b|x| sin(|x^3 + x|) from our exercise, as it requires differentiating the product of b|x| and sin(|x^3 + x|). Without the product rule, finding f'(x), and thus determining the differentiability at specific points, becomes significantly more challenging.

Limit of Derivative

To determine whether a function is differentiable at a particular point, we often examine the limit of its derivative as the variable approaches that point. If the limit exists and is the same from both the left and the right, the function is considered to be differentiable at that point. However, if the limits do not match or if even one of them doesn't exist, the function is not differentiable there.

The concept of a limit is crucial because it addresses the behavior of the function around the point of interest, and it's not sufficient for the function to merely have a derivative. For example, the limit of f'(x) as x approaches 0 is key to answering whether the given function f(x) in our exercise is differentiable at x=0. This is particularly pertinent when dealing with functions that have absolute values, as their derivatives can be undefined or discontinuous at certain points.

Absolute Value Function Differentiation

Absolute value functions, while straightforward in their definition, can be tricky when differentiation is involved due to their piecewise nature. For a function f(x) = |x|, the derivative is 1 for x > 0 and -1 for x < 0. However, the derivative does not exist at x = 0 because the graph of f(x) has a sharp corner at that point.

To differentiate functions that include absolute values, one often has to consider separate cases for positive and negative values of the variable within the absolute value. In the context of the given problem, the function f(x) has terms with absolute values, and so its differentiation would require careful application of piecewise differentiation. This consideration informs whether the given function is differentiable at certain values of x, such as 0 or 1, as required in the exercise.