Second derivatives, concavity and curve sketching: VCE Specialist Mathematics Units 3 & 4

1. The derivative has a rate of change too

The first derivative f′ tells you how the function's value changes as its input increases. The second derivative f″ tells you how that first derivative changes. These answer different questions: a graph can be increasing while its gradient is decreasing. For example, ln x rises for positive x, but becomes progressively flatter. Its first derivative is positive and its second derivative is negative.

Equivalent notation includes y″, f″(x) and d²y/dx². The last expression means differentiating twice with respect to x. It is not the square of dy/dx, and it is not obtained by separately squaring a numerator and denominator. If y is measured in metres and x in seconds, y′ has units m/s and y″ has units m/s².

To calculate f″, first find a correct first derivative, simplify if useful, and differentiate again. Use product, quotient and chain rules afresh. A simpler first-derivative form often reduces errors in the second calculation. Check that both expressions remain defined on the interval being analysed. Domain exclusions from the original function remain in force even when a derivative appears to simplify.

2. Concavity describes changing slopes

On an interval where f″ > 0, f′ is increasing and the graph is concave up. On an interval where f″ < 0, f′ is decreasing and the graph is concave down. Imagine moving along the graph from left to right: increasing slope can mean negative slopes becoming less negative, passing through zero and becoming positive. Therefore concave up does not mean the graph is increasing everywhere.

A useful local picture is the relation to a tangent. For a suitably smooth concave-up graph, nearby points lie above its tangent; for a concave-down graph, they lie below. This picture explains why tangent-based numerical approximations can systematically underestimate or overestimate a curve over a short step. It also helps you distinguish bending from vertical position.

The four sign combinations are all possible. Positive f′ and positive f″ mean rising with increasingly steep slope. Positive f′ and negative f″ mean rising while flattening. Negative f′ and positive f″ mean falling while flattening. Negative f′ and negative f″ mean falling with increasingly negative slope. Make a sketch of each before trying to memorise a verbal rule.

Concavity is an interval property. A single numerical value of f″ supplies local information when the derivative behaves continuously nearby, but does not establish the shape over a whole requested domain. For polynomial and simple rational examples, factor f″ and analyse signs on intervals separated by its zeros and undefined inputs.

3. Classify a stationary point with justified evidence

A stationary point has f′(a) = 0 at an input a in the domain. If f′ changes from positive to negative there, the point is a local maximum. If it changes from negative to positive, it is a local minimum. If the derivative retains its sign, it is not a turning point. Use the function value f(a) to provide the full coordinates.

The second-derivative test can be a shorter classification when its conditions apply. At a stationary point, f″(a) > 0 implies a local minimum and f″(a) < 0 implies a local maximum, for the smooth functions considered here. If f″(a) = 0, the test is inconclusive. It does not prove that the point is an inflection, nor that no extremum exists.

Compare three functions at zero. For x⁴, both first and second derivatives vanish, yet zero is a minimum: nearby values are nonnegative and f′ changes from negative to positive. For −x⁴, zero is a maximum. For x³, zero is a stationary inflection: f′ = 3x² stays nonnegative while f″ = 6x changes sign. The same pair of derivative values at the point can lead to three different classifications.

A local extremum is relative to nearby allowed inputs; a global extremum is relative to the entire stated domain. On a closed interval, endpoints can provide the greatest or least value without being stationary. For a maximum/minimum question on a restricted interval, compare endpoint values with all valid interior candidates. Do not accidentally solve a local classification problem when the question asks for an absolute bound.

4. Inflection requires a genuine change of concavity

For a smooth function, zeros of f″ are candidates for inflection. Test signs on both sides. If f″ merely touches zero without changing sign, there is no inflection. The function x⁴ has f″ = 12x², positive on both sides of zero, so the concavity does not change. The equation f″ = 0 identifies where to investigate, not the final answer.

Also consider points where the second derivative is undefined but the function itself exists. Such points require direct local concavity reasoning; a rule that only solves f″ = 0 is not a universal definition. The standard polynomial and rational problems here can usually be decided with sign charts, but the conceptual requirement is always a change in concavity at a point on the graph.

A vertical asymptote is not an inflection point. For f(x) = 1/x, f″ = 2/x³ has opposite signs across zero, but f(0) does not exist. A removable hole is also not a point of inflection in the original function. Always intersect candidate inputs with the original domain before giving coordinates or classifications.

5. A complete curve investigation follows a sequence

Start with the function's domain, intercepts, symmetry and asymptotes. These establish the graph's framework. Next use f′ for stationary points and increasing/decreasing intervals. Then use f″ for concavity and inflections. Finally combine all the evidence in one sketch. When the graph is restricted to an interval, mark whether each endpoint is included.

Keep three types of input in your sign charts: derivative zeros, derivative singularities and original domain exclusions. They divide the line into intervals, but do not all represent points on the graph. Use one exact test value in each interval when factor signs are not immediate. If a factor has an even power, its sign usually does not change through its zero; an odd power usually changes sign.

Coordinates should be exact when possible. An inflection at x = √3 belongs at height f(√3), not at a rounded height read from a small plotting window. A smooth drawing should respect the gradient direction at each turning point and the correct side of each asymptote. A well-labelled graph is a summary of reasoning, not a replacement for it.

6. Use technology to challenge a claim

Ask a CAS to simplify your derivative and compare it algebraically with a second independently obtained form. Then graph f, f′ and f″ in separate aligned windows. The zero crossings of f″ should correspond to changes in the direction of f′, not necessarily to zeros of f′. A numerical plot can suggest the sign intervals but should not replace exact reasoning when the factors are available.

As a non-routine investigation, consider f_a(x) = x⁴ − ax². Its stationary equation is 2x(2x² − a) = 0. When a > 0, there are three stationary points; when a ≤ 0, only zero remains. At a = 0 the second-derivative test fails, yet zero is still a minimum. Varying a on a graph helps locate the structural change; factoring and sign reasoning explain why it happens.

Common errors include equating concavity with increasing behaviour, using f″ = 0 as a complete inflection test, losing an implicit chain-rule factor and calling an excluded input a critical point on the curve. Repair each by naming the property being tested and writing the evidence that actually establishes it.