If f(x) = |2x + 5| , then f(x) is:

The correct answer is Option (4) → Continuous and non-differentiable at \(x = \displaystyle\frac{-5}{2}\)

The expression 2x + 5 changes sign at \(x = -\frac{5}{2}\), so

    \( \displaystyle f(x) = \begin{cases} -(2x + 5), & x \lt -\frac{5}{2}\\ 2x + 5, & x \ge -\frac{5}{2}\end{cases} \)

Continuity: both pieces tend to 0 as \(x \to -\frac{5}{2}\), and \(f\left(-\frac{5}{2}\right) = 0\). So f is continuous there (the modulus of a continuous function is continuous).

Differentiability: the left and right derivatives at \(x = -\frac{5}{2}\) are the slopes of the two pieces:

    \(Lf' = -2\)
    \(Rf' = 2\)

They are unequal (the graph has a corner there), so f is not differentiable at \(x = -\frac{5}{2}\).

Hence f is continuous and non-differentiable at \(x = -\frac{5}{2}\).