tmua.me Extreme Set C Paper 1 · Question 20

For $a\geq0$ , let

f_a(x)=x^3+a x(1-x).

The trapezium rule is used to estimate

\int_0^1 f_a(x)\,\mathrm dx

with two strips whose boundaries are $x=0$ , $x=t$ and $x=1$ , where $0<t<1$ . The value of $t$ may be chosen separately for each $a$ .

Let $m$ and $M$ be the smallest and largest values of $a$ for which this estimate can be exact. What is $mM$ ?

3-\dfrac{\sqrt3}{2}

\dfrac94

3+\dfrac{\sqrt3}{2}

\dfrac{15-\sqrt3}{6}

3

\dfrac{21}{4}-2\sqrt3

examday