Show that $f( - x) = - f(x)$ .

Given that $f''(x) = \frac{{2ax({x^2} - 3)}}{{{{({x^2} + 1)}^3}}}$ , find the coordinates of all points of inflexion.

It is given that $\int {f(x){\rm{d}}x = \frac{a}{2}} \ln ({x^2} + 1) + C$ .

(i)     Find the area of the shaded region, giving your answer in the form $p\ln q$ .

(ii)    Find the value of $\int_4^8 {2f(x - 1){\rm{d}}x}$ .

METHOD 1

evidence of substituting $- x$ for $x$     (M1)

$f( - x) = \frac{{a( - x)}}{{{{( - x)}^2} + 1}}$     A1

$f( - x) = \frac{{ - ax}}{{{x^2} + 1}}$ $( = - f(x))$     AG     N0

METHOD 2

$y = - f(x)$ is reflection of $y = f(x)$ in x axis

and $y = f( - x)$ is reflection of $y = f(x)$ in y axis     (M1)

sketch showing these are the same     A1

$f( - x) = \frac{{ - ax}}{{{x^2} + 1}}$ $( = - f(x))$     AG     N0

[2 marks]

evidence of appropriate approach     (M1)

e.g. $f''(x) = 0$

to set the numerator equal to 0     (A1)

e.g. $2ax({x^2} - 3) = 0$ ; $({x^2} - 3) = 0$

(0, 0) , $\left( {\sqrt 3 ,\frac{{a\sqrt 3 }}{4}} \right)$ , $\left( { - \sqrt 3 , - \frac{{a\sqrt 3 }}{4}} \right)$ (accept $x = 0$ , $y = 0$ etc)      A1A1A1A1A1     N5

[7 marks]

(i) correct expression     A2

e.g. $\left[ {\frac{a}{2}\ln ({x^2} + 1)} \right]_3^7$ , $\frac{a}{2}\ln 50 - \frac{a}{2}\ln 10$ , $\frac{a}{2}(\ln 50 - \ln 10)$

area = $\frac{a}{2}\ln 5$     A1A1     N2

(ii) METHOD 1

recognizing the shift that does not change the area     (M1)

e.g. $\int_4^8 {f(x - 1){\rm{d}}x}  = \int_3^7 {f(x){\rm{d}}x}$ , $\frac{a}{2}\ln 5$

recognizing that the factor of 2 doubles the area     (M1)

e.g. $\int_4^8 {2f(x - 1){\rm{d}}x = } 2\int_4^8 {f(x - 1){\rm{d}}x}$ $\left( { = 2\int_3^7 {f(x){\rm{d}}x} } \right)$

$\int_4^8 {2f(x - 1){\rm{d}}x = a\ln 5}$ (i.e. $2 \times$ their answer to (c)(i))     A1     N3

METHOD 2

changing variable

let $w = x - 1$ , so $\frac{{{\rm{d}}w}}{{{\rm{d}}x}} = 1$

$2\int {f(w){\rm{d}}w = } \frac{{2a}}{2}\ln ({w^2} + 1) + c$     (M1)

substituting correct limits

e.g. $\left[ {a\ln \left[ {{{(x - 1)}^2} + 1} \right]} \right]_4^8$ , $\left[ {a\ln ({w^2} + 1)} \right]_3^7$ , $a\ln 50 - a\ln 10$     (M1)

$\int_4^8 {2f(x - 1){\rm{d}}x = a\ln 5}$     A1     N3

[7 marks]

Part (a) was achieved by some candidates, although brackets around the $- x$ were commonly neglected. Some attempted to show the relationship by substituting a specific value for x . This earned no marks as a general argument is required.

Although many recognized the requirement to set the second derivative to zero in (b), a majority neglected to give their answers as ordered pairs, only writing the x-coordinates. Some did not consider the negative root.

For those who found a correct expression in (c)(i), many finished by calculating $\ln 50 - \ln 10 = \ln 40$ . Few recognized that the translation did not change the area, although some factored the 2 from the integrand, appreciating that the area is double that in (c)(i).