Q.Find the maximum and minimum values of $f$, if any, of the function given by $f(x) = |x|,\ x \in \mathbb{R}$.

The key idea is that $|x|$ is differentiable everywhere except at $x=0$, and its minimum occurs at the point where the derivative changes sign.

Step 1: For $x > 0$, $f(x) = x$, so $f'(x) = 1 > 0$ (strictly increasing).

For $x < 0$, $f(x) = -x$, so $f'(x) = -1 < 0$ (strictly decreasing).

At $x = 0$, $f$ is not differentiable (sharp corner), but it is continuous.

Step 2: Since $f$ decreases on $(-\infty, 0]$ and increases on $[0, \infty)$, the only critical point (where derivative changes sign) is $x = 0$.

$f(0) = 0$, and for any $x \neq 0$, $|x| > 0$. Hence $0$ is the absolute minimum.

Step 3: As $x \to \pm\infty$, $|x| \to \infty$, so no finite maximum exists.

Concept and Intuition

The absolute value function is one of the simplest piecewise-defined functions. Its graph is two rays meeting at the origin: for $x \ge 0$, $f(x) = x$ (a line with slope $+1$), and for $x < 0$, $f(x) = -x$ (a line with slope $-1$). The key feature is the sharp corner at $x=0$ — the function is not differentiable there, but it is continuous everywhere.

When we talk about "maximum and minimum values" of a function on its entire domain (here, all real numbers), we are looking for global (absolute) extrema. A function can have:

• A global minimum at a point where $f(x)$ is the smallest value over the whole domain.
• A global maximum at a point where $f(x)$ is the largest value over the whole domain.

For $f(x) = |x|$, the value is always non-negative. The smallest possible value is $0$, achieved at $x=0$. But can it ever be the largest? No — because as you move farther from zero in either direction, $|x|$ keeps increasing without any upper bound.

Step-by-Step Solution

1. Understand the domain and range.

   The domain is $\mathbb{R}$, all real numbers. The range of $|x|$ is $[0, \infty)$ — every non-negative real number appears as an output.

2. Check for a global minimum.

   For any $x \in \mathbb{R}$, we have $|x| \ge 0$. The equality $|x| = 0$ holds exactly when $x = 0$.

   Therefore, $f(0) = 0$ is less than or equal to every other function value.

   So $0$ is the global minimum value, attained at $x = 0$.

3. Check for a global maximum.

   Suppose there were a global maximum value $M$. Then for all $x$, $|x| \le M$. But pick $x = M+1$ (if $M \ge 0$) or $x = 1$ (if $M < 0$, which is impossible since $|x| \ge 0$). Then $|M+1| = M+1 > M$, a contradiction.

   More simply: as $x \to \infty$, $|x| \to \infty$, so no finite upper bound exists.

   Hence, no global maximum exists.

4. Summarise the findings.

   • Minimum value: $0$ (at $x=0$).
   • Maximum value: none.

Method: Finding Global Extrema of a Function That Is Not Differentiable Everywhere

Applies whenever a function has a "corner" or other point of non-differentiability (like $|x|$, or any piecewise function built from absolute values), and you're asked for its global maximum/minimum.

Steps

Step 1: Do not rely on $f'(x)=0$ alone to search for extrema.

The standard "solve $f'(x)=0$" search only finds smooth turning points. A global extremum can also occur exactly at a point where the derivative fails to exist — so any non-differentiable point in the domain must be checked separately as a candidate.

Step 2: At each non-differentiable point, examine the function's behaviour on either side directly, not via the derivative.

Check whether the function values increase moving away from that point on both sides (making it a minimum), decrease on both sides (a maximum), or do neither.

Step 3: Use a direct inequality argument to confirm a bound and show it's attained.

For example, if you can show $f(x) \ge 0$ for every $x$ using the definition of the function, and also find one specific $x$ where $f(x)=0$, that immediately proves $0$ is the global minimum — no derivative needed.

Step 4: Check boundedness for the opposite extremum, exactly as for any other function.

If the function grows without bound as $x \to \pm\infty$, state plainly that no maximum (or minimum) exists on that side.

Common Mistakes

Mistake 1: Assuming a function can't have a minimum or maximum at a point where it isn't differentiable.

Why it's wrong: the standard "solve $f'(x)=0$" search only finds smooth extrema; it says nothing about points where the derivative doesn't exist, like the corner of $|x|$ at $x=0$. A global extremum can occur at such a point too — here $f(0)=0$ is genuinely the smallest value $|x|$ ever takes. Correct approach: check points of non-differentiability directly using the definition of maximum/minimum by comparing function values, not only by solving $f'(x)=0$.