Differentiation

5 methods · 4 questions

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To locate or control where a derivative vanishes, write the derivative in fully factored form so its zeros are read off directly, and engineer the factors when you need prescribed stationary points. The recurring expert move is to peel off a factor that can never be zero (an exponential, a positive power) so that the stationary points are exactly the roots of the remaining polynomial.

Trigger: "find where the derivative is zero", or "construct a function whose derivative vanishes at given points".

Instances:
(i) Differentiate $P (x) e^{- x^{2}}$ , factor out the never-zero $e^{- x^{2}}$ , and read the stationary points as the roots of the polynomial that remains.
(ii) Reverse the process: to force stationary points at $0, \pm a, \pm b$ , choose $P$ so the bracket factors as $x (x^{2} - a^{2}) (x^{2} - b^{2})$ up to the exponential's contribution.
(iii) For a cubic, the derivative is a quadratic whose two roots are the maximum and minimum abscissae, handed straight to Vieta.
(iv) For $x^{x}$ and its relatives, after logarithmic differentiation the stationary point is the single zero of the leftover factor $1 + ln x$ .

Linked questions (3)

STEP 2 2017 · Q7

The functions

f

and

g

are defined, for

x > 0

, by

f (x) = x^{x}, g (x) = x^{f (x)} .

By taking logarithms, or otherwise, show that $f (x) > x$ for $0 < x < 1$ . Show further that $x < g (x) < f (x)$ for $0 < x < 1$ . Write down the corresponding results for $x > 1$ .
Find the value of $x$ for which $f^{'} (x) = 0$ .
Use the result $x ln x \to 0$ as $x \to 0$ to find $x \to 0 lim f (x)$ , and write down $x \to 0 lim g (x)$ .
Show that $x^{- 1} + ln x ⩾ 1$ for $x > 0$ . Using this result, or otherwise, show that $g^{'} (x) > 0$ .

Sketch the graphs, for

x > 0

, of

y = x

, \

y = f (x)

and

y = g (x)

on the same axes.

STEP 3 2018 · Q1

The function $f$ is given by $f (β) = β - \frac{1}{β} - \frac{1}{β ^{2}} (β \neq = 0) .$ Find the stationary point of the curve $y = f (β)$ and sketch the curve. Sketch also the curve $y = g (β)$ , where $g (β) = β + \frac{3}{β} - \frac{1}{β ^{2}} (β \neq = 0) .$
Let $u$ and $v$ be the roots of the equation $x^{2} + α x + β = 0,$ where $β \neq = 0$ . Obtain expressions in terms of $α$ and $β$ for $u + v + \frac{1}{uv}$ and $\frac{1}{u} + \frac{1}{v} + uv$ .
Given that $u + v + \frac{1}{uv} = - 1$ , and that $u$ and $v$ are real, show that $\frac{1}{u} + \frac{1}{v} + uv ⩽ - 1$ .
Given instead that $u + v + \frac{1}{uv} = 3$ , and that $u$ and $v$ are real, find the greatest value of $\frac{1}{u} + \frac{1}{v} + uv$ .

STEP 3 2022 · Q3

(i) The curve

C_{1}

has equation

a x^{2} + b x y + c y^{2} = 1

where

ab c \neq = 0

and

a > 0

.

Show that, if the curve has two stationary points, then

b^{2} < 4 a c

.

(ii) The curve

C_{2}

has equation

a y^{3} + b x^{2} y + c x = 1

where

ab c \neq = 0

and

b > 0

.

Show that the

x

-coordinates of stationary points on this curve satisfy

4 c b^{3} x^{4} - 8 b^{3} x^{3} - a c^{3} = 0.

Show that, if the curve has two stationary points, then

4 a c^{6} + 27 b^{3} > 0

.

(iii) Consider the simultaneous equations

a y^{3} + b x^{2} y + c x = 1

2 b x y + c = 0

3 a y^{2} + b x^{2} = 0

where

ab c \neq = 0

and

b > 0

.

Show that, if these simultaneous equations have a solution, then

4 a c^{6} + 27 b^{3} = 0