Derivatives measure the instantaneous slope of a function. This chapter turns that local information into global statements: where a function reaches its largest or smallest values, how the sign of the derivative controls whether a function rises or falls, how the sign of the second derivative controls the shape of its graph, and, tying all of it together, the Mean Value Theorem, which guarantees that a function's average rate of change over an interval is realized as an instantaneous rate of change somewhere inside it.
Absolute and Local Extrema
For \(f:D\to\mathbb{R}\), a point \(c\in D\) is an absolute (global) maximum if \(f(c)\ge f(x)\) for every \(x\in D\), and an absolute minimum if \(f(c)\le f(x)\) for every \(x\in D\). These compare \(f(c)\) against the entire domain at once.
A point \(c\) is a local maximum if \(f(c)\ge f(x)\) for every \(x\) in some open interval around \(c\), and a local minimum if \(f(c)\le f(x)\) there. A local extremum only needs to beat its immediate neighbors; it need not be the largest or smallest value anywhere else in \(D\).
Critical Points and Fermat's Theorem
A point \(c\) in the interior of \(D\) is a critical point of \(f\) if \(f'(c)=0\) or \(f'(c)\) does not exist. Critical points are exactly the candidates for interior local extrema:
\[ \boxed{ f\text{ has a local extremum at an interior point }c\text{ where }f\text{ is differentiable} \implies f'(c)=0 } \]
This is Fermat's theorem. It follows directly from the sign of the difference quotient: if \(f\) has, say, a local maximum at \(c\), then \(f(x)-f(c)\le0\) for \(x\) near \(c\). For \(x>c\), dividing by the positive \(x-c\) keeps the sign, giving \(\frac{f(x)-f(c)}{x-c}\le0\) in the limit, so \(f'(c)\le0\); for \(x<c\), dividing by the negative \(x-c\) flips it, giving \(f'(c)\ge0\). Both hold only if \(f'(c)=0\).
Fermat's theorem is a necessary, not a sufficient, condition. As already seen with \(f(x)=x^3\) in Derivatives: \(f'(0)=0\), yet \(x=0\) is neither a local maximum nor a local minimum, since \(f\) is negative just left of \(0\) and positive just right of it.
The Extreme Value Theorem
Critical points only account for interior behavior; a function can also attain its extrema at the boundary of its domain. On a closed, bounded interval, both possibilities are guaranteed to actually occur:
\[ \boxed{ f\text{ continuous on }[a,b] \implies f\text{ attains both an absolute maximum and an absolute minimum on }[a,b] } \]
Both hypotheses are essential: \(f(x)=x\) on the open interval \((0,1)\) attains neither endpoint value, and \(f(x)=1/x\) on \((0,1]\) is unbounded near \(0\); both fail because the interval is not closed. Similarly, a continuous function need not attain its extrema if the domain is closed but unbounded, such as \(f(x)=x\) on \([0,\infty)\).
The Closed Interval Method
Together, Fermat's theorem and the Extreme Value Theorem give a complete procedure for locating the absolute extrema of a continuous \(f\) on \([a,b]\): the maximum and minimum must occur either at a critical point in \((a,b)\) or at one of the two endpoints, so it suffices to evaluate \(f\) at all of these finitely many candidates and compare.
For \(f(x)=x^3-3x\) on \([-2,2]\), the derivative \(f'(x)=3x^2-3\) vanishes at \(x=\pm1\), both interior to the interval. Evaluating \(f\) at the two critical points and the two endpoints,
\[ f(-2)=-2,\qquad f(-1)=2,\qquad f(1)=-2,\qquad f(2)=2, \]
the largest value \(2\) is attained at both \(x=-1\) and \(x=2\), and the smallest value \(-2\) at both \(x=-2\) and \(x=1\) — a reminder that absolute extrema need not be unique.
Monotonicity and the First Derivative Test
As already noted when the derivative was first introduced, \(f'(x)>0\) on an interval makes \(f\) strictly increasing there, and \(f'(x)<0\) makes it strictly decreasing; where \(f'(x)=0\) throughout an interval, \(f\) is constant. This turns the sign of \(f'\) around a critical point into a direct classification of that point:
| Sign change of \(f'\) at \(c\) | Behavior at \(c\) |
|---|---|
| \(+\to-\) | local maximum |
| \(-\to+\) | local minimum |
| no change (\(+\to+\) or \(-\to-\)) | no local extremum |
For \(f(x)=x^3-3x^2\), \(f'(x)=3x^2-6x=3x(x-2)\) has roots at \(x=0\) and \(x=2\). Checking the sign of \(f'\) on the three resulting intervals,
\[ f'(x)>0\text{ on }(-\infty,0), \qquad f'(x)<0\text{ on }(0,2), \qquad f'(x)>0\text{ on }(2,\infty), \]
the sign flips from \(+\) to \(-\) at \(x=0\), giving a local maximum \(f(0)=0\), and from \(-\) to \(+\) at \(x=2\), giving a local minimum \(f(2)=-4\).
Concavity, Inflection Points, and the Second Derivative Test
The second derivative \(f''\) describes how the slope itself changes. Where \(f''(x)>0\), the slope is increasing and the graph curves upward (concave up, or convex); where \(f''(x)<0\), the slope is decreasing and the graph curves downward (concave down, or concave). A point where the concavity actually switches sign is called an inflection point; \(f''(c)=0\) alone is only necessary for one, since the sign of \(f''\) must genuinely change across \(c\), not merely touch zero.
For \(f(x)=x^3\), \(f''(x)=6x\) is negative for \(x<0\) and positive for \(x>0\), so \((0,0)\) is an inflection point even though, as seen above, it is not a local extremum.
At a critical point where \(f'(c)=0\), the concavity directly decides which kind of extremum it is, without needing to check the sign of \(f'\) on either side:
\[ f''(c)>0\implies\text{local minimum}, \qquad f''(c)<0\implies\text{local maximum}, \qquad f''(c)=0\implies\text{inconclusive}. \]
Intuitively, an upward-curving graph around a horizontal tangent sits above that tangent locally, which is exactly the shape of a trough; a downward-curving one sits below it, the shape of a peak. When \(f''(c)=0\), the test gives no information and the first derivative test (or a higher-order check) is needed instead.
Revisiting \(f(x)=x^3-3x^2\) with \(f''(x)=6x-6\) confirms the first derivative test's conclusions from a different angle: \(f''(0)=-6<0\) gives a local maximum at \(x=0\), and \(f''(2)=6>0\) gives a local minimum at \(x=2\).
The Mean Value Theorem
Fermat's theorem related an extremum to a horizontal tangent. The Mean Value Theorem generalizes this to tangents parallel to any secant line:
\[ \boxed{ f\text{ continuous on }[a,b]\text{ and differentiable on }(a,b) \implies \exists\,c\in(a,b):\ f'(c)=\frac{f(b)-f(a)}{b-a} } \]
The right-hand side is the average rate of change of \(f\) over \([a,b]\); the theorem asserts that this average is realized as an actual instantaneous rate of change at some interior point. Geometrically, the tangent line at \(c\) is parallel to the secant line through \((a,f(a))\) and \((b,f(b))\).
For \(f(x)=x^2\) on \([1,3]\), the average rate of change is
\[ \frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4. \]
Since \(f'(x)=2x\), solving \(2c=4\) gives \(c=2\), which indeed lies in \((1,3)\): the tangent at \(x=2\) has the same slope as the secant across the whole interval.
The Mean Value Theorem is not just a curiosity: it is exactly the tool used, for instance, to justify replacing the derivative at an unknown point with the average slope of a chord when deriving the arc length of a function, and it underlies why a function with \(f'\equiv0\) on an interval must be constant there.
Rolle's Theorem
When the two endpoint values coincide, the secant is horizontal, so the guaranteed tangent is horizontal too. This special case is usually stated on its own as Rolle's theorem:
\[ \boxed{ f\text{ continuous on }[a,b],\ \text{differentiable on }(a,b),\ f(a)=f(b) \implies \exists\,c\in(a,b):\ f'(c)=0 } \]
This is immediate from the Mean Value Theorem with \(f(b)-f(a)=0\): between any two points where a differentiable function returns to the same value, there is at least one point with a horizontal tangent.