Question

The function $f(x)=|x^{2}+x-6|$ is not differentiable at $x=a$ and $x=b$ then $(b-a)^{2}$ equals

• A. 25
• B. 9
• C. 13
• D. 5

Hint:

Points where a function is not differentiable are often the points where the function touches or crosses the x-axis inside an absolute value sign.

Answer 1

The Correct Option is A

A function $|g(x)|$ is generally not differentiable at the roots of $g(x)$.
Step 1: Let $g(x) = x^2 + x - 6$. The absolute value function creates "sharp points" or "corners" at the x-axis where the polynomial changes sign.
Step 2: We find the roots of $x^2 + x - 6 = 0$. $x^2 + 3x - 2x - 6 = 0$. $x(x + 3) - 2(x + 3) = 0$. $(x - 2)(x + 3) = 0$. The roots are $x = 2$ and $x = -3$.
Step 3: The function $f(x)$ is not differentiable at its roots because the derivative from the left and right will have opposite signs. Thus, $a = -3$ and $b = 2$.
Step 4: We need to find $(b - a)^2$: $(b - a) = 2 - (-3) = 5$. $(b - a)^2 = 5^2 = 25$. The result is 25, which matches Option (1).