6.4 Maxima-Minima, Indeterminate Forms & Taylor Series
Key Takeaways
- At a stationary point f'(x) = 0, a positive second derivative indicates a local minimum, a negative one a local maximum, and a zero second derivative demands a higher-order or sign-change test.
- L'Hopital's rule applies only to the indeterminate forms 0/0 and infinity/infinity; other indeterminate forms must first be algebraically converted into one of these two.
- The Taylor series expands f(x) about x = a using successive derivatives divided by factorials, and the Maclaurin series is simply the Taylor series about a = 0.
- Truncating a Taylor series after the linear term is the mathematical basis for small-angle approximations and for linearising governor and vibration equations later in Paper-II.
Maxima and Minima of a Single Variable
Optimisation is the most directly useful calculus topic for a mechanical engineer: minimum cost of machining, maximum power transmitted by a belt, optimum order quantity in inventory control. All of them follow the same two-step recipe.
Step 1 — locate stationary points. Solve $f'(x) = 0$.
Step 2 — classify them using the second derivative test:
| $f''(x)$ at the stationary point | Nature |
|---|---|
| $f''(x) > 0$ | Local minimum (curve concave up) |
| $f''(x) < 0$ | Local maximum (curve concave down) |
| $f''(x) = 0$ | Test fails — check higher derivatives or sign change |
When $f''(x) = 0$ the point may be a maximum, a minimum, or a point of inflection. The reliable fallback is the first-derivative sign test: if $f'$ changes from positive to negative, it is a maximum; negative to positive, a minimum; no change, an inflection.
For a closed interval $[a,b]$, the absolute extremum must be found by comparing the values at all interior stationary points and at both endpoints. Candidates routinely forget the endpoints.
Worked example. A rectangular settling tank of fixed volume $V = 32$ m$^3$ has a square base of side $x$ and height $h$, and is open at the top. Minimise the surface area.
With $h = 32/x^2$, the area is
Then $dA/dx = 2x - 128/x^2 = 0$ gives $x^3 = 64$, so $x = 4$ m and $h = 2$ m. Check: $d^2A/dx^2 = 2 + 256/x^3 = 6 > 0$, confirming a minimum. The minimum area is $16 + 32 = 48$ m$^2$.
Indeterminate Forms
Seven forms are indeterminate:
L'Hopital's rule applies directly only to the first two. If $f(a) = g(a) = 0$ (or both tend to infinity) and $g'(a) \neq 0$, then
and the rule may be applied repeatedly while the form remains indeterminate. Applying it to a form that is not indeterminate — for instance $2/0$ — produces a wrong answer, and this is the single most common error on such items.
The other five forms are converted first. A product $0 \times \infty$ is rewritten as a quotient; a difference $\infty - \infty$ is combined over a common denominator; the three exponential forms are handled by taking logarithms.
Worked example. $\lim_{x\to0} \dfrac{e^x - 1 - x}{x^2}$ is of the form $0/0$. Differentiating gives $\dfrac{e^x - 1}{2x}$, still $0/0$. Differentiating again gives $\dfrac{e^x}{2} \to \dfrac{1}{2}$.
Taylor and Maclaurin Series
Any function with sufficiently many derivatives at $x = a$ can be expanded as
Setting $a = 0$ gives the Maclaurin series
The expansions you must recall without derivation:
Note the parity: $\sin x$ contains only odd powers, $\cos x$ only even powers. A question offering an even-power term in the sine expansion can be eliminated on inspection.
Why This Matters Downstream in Paper-II
Truncating these series after the first-order term is the formal justification for approximations you will use repeatedly:
- $\sin\theta \approx \theta$ and $\cos\theta \approx 1$ for small $\theta$ — used to linearise the simple pendulum and the governor equations in Theory of Machines.
- $(1+x)^n \approx 1 + nx$ — used for small changes in efficiency, density and clearance calculations.
- The two-term expansion of $e^x$ underlies the linearisation of exponential temperature decay in lumped-capacitance heat transfer.
A Taylor expansion truncated after the linear term is exactly a tangent-line approximation, which connects back to the mean value theorem of the previous section.
Worked Example: Approximating a Root
Estimate $\sqrt{4.1}$ using a two-term expansion. Write $\sqrt{4.1} = 2\sqrt{1 + 0.025}$ and apply $(1+x)^{1/2} \approx 1 + x/2$:
The true value is 2.02485, so the linear term alone gives four-figure accuracy — which is why this technique is worth more than a calculator in a timed paper.
At a stationary point of f(x), the second derivative is found to be negative. The point is a:
L'Hopital's rule may be applied directly only when the limit takes which form?
In the Maclaurin expansion of sin x, which powers of x appear?
An open-top tank with a square base of side x and fixed volume 32 cubic metres has surface area A = x^2 + 128/x. The side length that minimises the surface area is: