Question

If \( f(x) = |x^2 + x - 6| \) is not differentiable at \( x = a \) and \( x = b \), then \( a^2 + b^2 = \)

• A. \(11 \)
• B. \(14 \)
• C. \(12 \)
• D. \(13 \)
• E. \(16 \)

Hint:

Modulus functions are non-differentiable at points where inside expression = 0.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A function involving an absolute value, \(f(x) = |g(x)|\), is not differentiable at the points where the expression inside the absolute value, \(g(x)\), is equal to zero and has a simple root (i.e., the graph of \(g(x)\) crosses the x-axis, creating a "sharp corner" in the graph of \(f(x)\)).
Step 2: Key Formula or Approach:
1. Set the expression inside the absolute value to zero to find the potential points of non-differentiability. 2. The function is \(g(x) = x^2 + x - 6\). We need to solve the equation \(x^2 + x - 6 = 0\). 3. The roots of this equation will be the values `a` and `b`. 4. Calculate \(a^2 + b^2\).
Step 3: Detailed Explanation:
The function is \(f(x) = |x^2 + x - 6|\). The points of non-differentiability occur where the argument of the absolute value function is zero. So, we set: \[ x^2 + x - 6 = 0 \] We can solve this quadratic equation by factoring. We need two numbers that multiply to -6 and add to 1. These numbers are 3 and -2. \[ (x + 3)(x - 2) = 0 \] The roots are: \[ x = -3 \quad \text{and} \quad x = 2 \] These are the points `a` and `b` where the function is not differentiable. Let \(a = -3\) and \(b = 2\). Now, we need to calculate \(a^2 + b^2\): \[ a^2 + b^2 = (-3)^2 + (2)^2 \] \[ a^2 + b^2 = 9 + 4 = 13 \] Step 4: Final Answer:
The value of \(a^2 + b^2\) is 13.