Question

The function \(f(x) = |x^2 - 2x - 3| \cdot e^{|9x^2 - 12x + 4|}\) is not differentiable at exactly :

• A. one point
• B. two points
• C. three points
• D. four points

Hint:

For \(|f(x)|\), check if the roots of \(f(x)\) are multiple roots. If a root is even-ordered (like a perfect square), the absolute value doesn't cause a cusp, and it remains differentiable.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A function of the form \(|g(x)| \cdot h(x)\) is not differentiable at points where \(g(x) = 0\), provided \(g'(x) \neq 0\) at those points and \(h(x)\) is smooth.
Step 2: Detailed Explanation:
Let's analyze the absolute value terms:
1. First term: \(x^2 - 2x - 3 = (x - 3)(x + 1)\). The roots are \(x = 3\) and \(x = -1\). The derivative of the inner function is \(2x - 2\), which is non-zero at both roots (\(2(3)-2 = 4\) and \(2(-1)-2 = -4\)). Thus, \(|x^2 - 2x - 3|\) is non-differentiable at \(x = 3\) and \(x = -1\).
2. Second term: \(9x^2 - 12x + 4 = (3x - 2)^2\). Since this is a perfect square, it is always non-negative.
Therefore, \(|9x^2 - 12x + 4| = |(3x - 2)^2| = (3x - 2)^2\).
The function becomes \(e^{(3x - 2)^2}\), which is a smooth composition of smooth functions and is differentiable for all real \(x\).
3. Full function: \(f(x) = |x^2 - 2x - 3| \cdot e^{(3x - 2)^2}\). The product of a non-differentiable function and a non-zero smooth function is non-differentiable.
Since \(e^{(3x - 2)^2}\) is never zero, the points of non-differentiability of \(f(x)\) are exactly the points where \(|x^2 - 2x - 3|\) is not differentiable.
Step 3: Final Answer:
The function is not differentiable at exactly two points: \(x = -1\) and \(x = 3\).