Question

The number of points, at which the function f(x) = |2x + 1| - 3|x + 2| + |x² + x - 2|, x ∈ ℝ is not differentiable, is ________

Hint:

A function $|g(x)|$ is generally not differentiable where $g(x)=0$, unless $g(x)$ has a repeated root at that point.

Answer 1

Correct Answer: 2

Step 1: Critical points for $|2x+1|$ is $x = -1/2$.
Step 2: Critical points for $|x+2|$ is $x = -2$.
Step 3: Critical points for $|x^2+x-2| = |(x+2)(x-1)|$ are $x = -2$ and $x = 1$.
Step 4: Potential points of non-differentiability: $\{-2, -1/2, 1\}$.
Step 5: At $x = -2$, the term $|x+2|$ appears twice (once in the linear part, once in the quadratic). We must check if they "smooth" each other out.
Step 6: Near $x=-2$, $f(x) \approx ....... -3|x+2| + |x+2||x-1| = |x+2|(-3 + |x-1|)$. Since $(-3+|x-1|) \to -3+3=0$ at $x=-2$, the function actually becomes differentiable at $x=-2$!
Step 7: Points remaining: $\{-1/2, 1\}$. Total = 2. \itextbf{*Correction based on rigorous check: $x=-2$ is a cusp because the slope changes from $-3(-1) + (-1)(3) = 0$ to something else. Re-evaluating: points are $-2, -1/2, 1$. Count = 3.*}