Finding Points of Non-Differentiability

A function of the form $f(x) = |g(x)|$ is not differentiable at points where $g(x) = 0$ and $g'(x) \neq 0$.

In this case, $g(x) = x^2 + x - 6$. We need to find the roots of $g(x)$.

Step 1: Find roots of $g(x)$

Set the quadratic expression to zero:

$ x^2 + x - 6 = 0 $

Factor the quadratic:

$ (x+3)(x-2) = 0 $

The roots are $x = -3$ and $x = 2$. These are the potential points of non-differentiability.

Step 2: Check the derivative $g'(x)$ at the roots

Calculate the derivative of $g(x)$:

$ g'(x) = \frac{d}{dx}(x^2 + x - 6) = 2x + 1 $

Evaluate $g'(x)$ at the roots:

• At $x = -3$: $g'(-3) = 2(-3) + 1 = -6 + 1 = -5$. Since $g'(-3) \neq 0$, the function is not differentiable at $x = -3$.
• At $x = 2$: $g'(2) = 2(2) + 1 = 4 + 1 = 5$. Since $g'(2) \neq 0$, the function is not differentiable at $x = 2$.

The points where the function $f(x)$ is not differentiable are $a = -3$ and $b = 2$ (or vice versa).

Calculating $(b-a)^2$

We need to calculate the value of $(b-a)^2$ using the points $a = -3$ and $b = 2$.

Step 3: Calculate the difference $(b-a)$

$ b - a = 2 - (-3) = 2 + 3 = 5 $

Step 4: Square the difference

$ (b-a)^2 = (5)^2 = 25 $

Thus, the value of $(b-a)^2$ is 25.