What Derivative Data Tell Us
A derivative sign chart is a compact description of some features of a curve. The sign of the first derivative indicates whether a function is increasing or decreasing on suitable intervals. The sign of the second derivative indicates concavity on suitable intervals.
But a sign chart is not a complete description of a function. It does not usually specify exact heights, slopes, intercepts, or scales. It also cannot settle behavior across a point where the relevant derivative is undefined unless additional information is available.
What a First-Derivative Sign Pattern Determines
Let \(f\) be differentiable on an interval \(I\). If \(f'(x)>0\) for every \(x\in I\), then \(f\) is strictly increasing on \(I\). If \(f'(x)<0\) throughout \(I\), then \(f\) is strictly decreasing there. If \(f'(x)\ge 0\) throughout \(I\), then \(f\) is nondecreasing; if \(f'(x)\le 0\), it is nonincreasing.
These conclusions follow from the Mean Value Theorem. For any \(a<b\) in \(I\), differentiability on the interval ensures continuity on \([a,b]\) and differentiability on \((a,b)\). The theorem gives a point \(c\in(a,b)\) such that \(f(b)-f(a)=f'(c)(b-a)\). Since \(b-a>0\), the sign of the difference is controlled by the sign of \(f'(c)\).
Consequently, if two functions are differentiable throughout the same interval and their first derivatives have the same strictly positive or strictly negative sign there, they have the same increasing or decreasing behavior there. This does not say that their slopes have the same sizes, or that their values agree.
Worked Example: Same Increase, Different Heights and Slopes
Compare \(f(x)=2x-1\) and \(g(x)=5x+4\) on \(\mathbb R\). Their derivatives are \(f'(x)=2\) and \(g'(x)=5\), both positive for every real \(x\). Thus both functions are strictly increasing on \(\mathbb R\).
Their heights and slopes differ. At \(x=0\), \(f(0)=2(0)-1=-1\), while \(g(0)=5(0)+4=4\). The slopes are \(2\) and \(5\), respectively. They also have different zeros: solving \(2x-1=0\) gives \(x=\frac12\); solving \(5x+4=0\) gives \(x=-\frac45\).
Check: \(f'(x)=2>0\) and \(g'(x)=5>0\), confirming that the shared feature is increasing behavior, not equal height, slope, or intercept.
What a Second-Derivative Sign Pattern Determines
If \(f\) is twice differentiable throughout an interval \(I\), then \(f''(x)>0\) throughout \(I\) means that \(f\) is concave up there, and \(f''(x)<0\) throughout \(I\) means that it is concave down there. A second-derivative sign pattern therefore constrains the way the slope changes, but it does not determine the function’s vertical position or, by itself, whether the function is increasing or decreasing.
Worked Example: Shared Concavity and Turning Location, Different Minimum Values
Let \(f(x)=(x-2)^2\) and \(g(x)=3(x-2)^2+5\). Their first derivatives are \(f'(x)=2(x-2)\) and \(g'(x)=6(x-2)\). Each derivative is negative for \(x<2\), zero at \(x=2\), and positive for \(x>2\). Both functions decrease to \(x=2\) and increase after it.
The second derivatives are \(f''(x)=2\) and \(g''(x)=6\), so both functions are concave up on all of \(\mathbb R\). At \(x=2\), their values are \(f(2)=0^2=0\) and \(g(2)=3(0)^2+5=5\). Their minimum values are different.
For a direct check of the vertical difference, \(f(3)=(3-2)^2=1\), whereas \(g(3)=3(3-2)^2+5=8\). Differentiating the displayed functions gives the first and second derivatives above, confirming the shared sign patterns and different scales and heights.
Sign Data Are Not the Same as Derivative Values
A sign pattern records whether a derivative is positive, negative, or zero; it does not record its exact value. Even exact derivative values leave some freedom: if two differentiable functions have the same first derivative on an interval, their difference is constant there. Thus their graphs can differ by a vertical shift.
For the first claim, take any \(a<b\) in \(I\). The Mean Value Theorem applied to \(h=f-g\) gives \(h(b)-h(a)=h'(c)(b-a)\) for some \(c\in(a,b)\). Since \(h'(c)=f'(c)-g'(c)=0\), it follows that \(h(b)=h(a)\). As this holds for any pair \(a,b\), \(h\) is constant.
For the second claim, \(h=f-g\) satisfies \(h''=0\). Applying the first claim to \(h'\) shows that \(h'\) is constant, say \(A\). Applying it again to \(h(x)-Ax\), whose derivative is zero, shows \(h(x)-Ax=B\) for a constant \(B\). Hence \(f(x)-g(x)=Ax+B\). These claims apply on intervals where the stated derivatives exist throughout; they do not automatically connect separate domain intervals across a break.
Worked Example: Equal Second Derivatives, Different First-Derivative Behavior
Compare \(f(x)=x^2\) and \(h(x)=x^2+4x+1\). Their second derivatives are \(f''(x)=2\) and \(h''(x)=2\), so both are concave up everywhere. But \(f'(x)=2x\) changes sign at \(x=0\), while \(h'(x)=2x+4=2(x+2)\) changes sign at \(x=-2\). Their minima occur at different locations.
Completing the square gives \(h(x)=(x+2)^2-3\), so \(h(-2)=-3\). Also \(f(0)=0\). Their difference is \(h(x)-f(x)=4x+1\), an affine function, as the theorem predicts.
Check: differentiating \(x^2+4x+1\) gives \(2x+4\), and differentiating again gives \(2\), matching \(f''(x)=2\). Equal second derivatives fix the concavity pattern here, but not the first-derivative sign-change location or the minimum value.
Why Undefined Derivatives Need Separate Attention
A sign chart may divide the domain at a point where a derivative is undefined. Matching signs on the intervals to either side does not, on its own, establish how the function behaves across that point. In particular, the Mean Value Theorem cannot be used across a discontinuity: its continuity requirement on the closed interval is essential.
Worked Example: Positive Derivatives on Either Side Do Not Ensure Increase Across a Break
Define \(f(x)=2x\) for \(x<0\) and \(f(x)=x\) for \(x\ge0\). Define \(g(x)=2x\) for \(x<0\) and \(g(x)=x-1\) for \(x\ge0\). For both functions, the derivative is \(2\) when \(x<0\) and \(1\) when \(x>0\); each derivative is undefined at \(0\).
The function \(f\) is increasing across \(0\): values to the left are negative, \(f(0)=0\), and values to the right are nonnegative; within each side the formulas have positive slopes. But \(g(-0.1)=2(-0.1)=-0.2\), while \(g(0)=0-1=-1\). Since \(-0.2>-1\) even though \(-0.1<0\), \(g\) is not increasing across \(0\).
Check: on each open side both functions have positive derivatives, but \(g\) has a jump at \(0\). The matching derivative signs on those sides do not determine the comparison across the break.
There is a corresponding caution for second derivatives. For \(x\ne0\), let \(F(x)=x^2\) when \(x<0\) and \(F(x)=x^2+x\) when \(x\ge0\); let \(G(x)=x^2\) when \(x<0\) and \(G(x)=x^2-x\) when \(x\ge0\). Both have second derivative \(2\) on either open side of \(0\), and their second derivatives are undefined at \(0\). Yet \(F\) is concave up across \(0\), while \(G\) is not. To see the failure for \(G\), take \(a=-0.1\) and \(b=0.1\). Then \(G(a)=0.01\), \(G(b)=0.01-0.1=-0.09\), and their average is \(-0.04\), while \(G(0)=0\). A concave-up function must lie at or below the chord midpoint, but \(0\) is not at or below \(-0.04\). Therefore the second-derivative signs away from the break do not settle concavity across it.
What the Patterns Fix—and What They Leave Open
The useful distinction is between a qualitative constraint and a complete reconstruction. Under the differentiability conditions stated above, matching first-derivative signs gives matching increasing or decreasing behavior on each chart interval. Matching second-derivative signs gives matching concavity on each interval where both functions are twice differentiable. Sign changes can help locate candidate features, but a sign chart alone does not provide exact function values.
In particular, derivative signs do not generally determine:
- the vertical position or \(y\)-intercept of a curve;
- the numerical slope at a point, or the scale of the graph;
- the exact function values at extrema or elsewhere;
- behavior across a point where the relevant derivative is undefined.
For example, adding a constant \(C\) to a function leaves both derivatives unchanged: \((f+C)'=f'\) and \((f+C)''=f''\). Multiplying by a positive constant \(a\) preserves the signs of both derivatives, because \((af)'=af'\) and \((af)''=af''\). Thus vertical shifts and positive changes of scale can preserve derivative sign patterns while changing the graph.
Check Your Understanding
Use the derivative information and the conditions on the relevant intervals to answer these questions.
- If two differentiable functions have positive first derivatives throughout the same interval, what can you conclude about their monotonicity? Name one feature you cannot conclude is the same.
- If two twice-differentiable functions have the same positive second derivative throughout an interval, what shared property follows? Does that information alone determine where either function decreases?
- If \(f'=g'\) throughout an interval, what form must \(f-g\) have? If instead \(f''=g''\), what form can \(f-g\) have?
- Why is it insufficient to know that a derivative is positive on each side of a point where it is undefined when deciding whether a function increases across the point?
- What additional issue arises when using second-derivative signs to discuss concavity across a point where the second derivative is undefined?
