Find the exact values of $a$ and $b$ if $f$ is continuous and differentiable at $x = 1$.

(i)     Use Rolle’s theorem, applied to $f$, to prove that $2{x^4} - 4{x^3} - 5{x^2} + 4x + 1 = 0$ has a root in the interval $\left] { - 1,1} \right[$.

(ii)     Hence prove that $2{x^4} - 4{x^3} - 5{x^2} + 4x + 1 = 0$ has at least two roots in the interval $\left] { - 1,1} \right[$.

$\mathop {{\text{lim}}}\limits_{x \to {1^ - }} {{\text{e}}^{ - {x^2}}}\left( { - {x^3} + 2{x^2} + x} \right) = \mathop {{\text{lim}}}\limits_{x \to {1^ + }}  (ax + b)$   $( = a + b)$     M1

$2{{\text{e}}^{ - 1}} = a + b$     A1

differentiability: attempt to differentiate both expressions     M1

$f'(x) =  - 2x{{\text{e}}^{ - {x^2}}}\left( { - {x^3} + 2{x^2} + x} \right) + {{\text{e}}^{ - {x^2}}}\left( { - 3{x^2} + 4x + 1} \right)$   $(x < 1)$     A1

(or $f'(x) = {{\text{e}}^{ - {x^2}}}\left( {2{x^4} - 4{x^3} - 5{x^2} + 4x + 1} \right)$)

$f'(x) = a$   $(x > 1)$     A1

substitute $x = 1$ in both expressions and equate

$- 2{{\text{e}}^{ - 1}} = a$     A1

substitute value of $a$ and find $b = 4{{\text{e}}^{ - 1}}$     M1A1

[8 marks]

(i)     $f'(x) = {{\text{e}}^{ - {x^2}}}\left( {2{x^4} - 4{x^3} - 5{x^2} + 4x + 1} \right)$   (for $x \leqslant 1$)     M1

$f(1) = f( - 1)$     M1

Rolle’s theorem statement     (A1)

by Rolle’s Theorem, $f'(x)$ has a zero in $\left] { - 1,1} \right[$     R1

hence quartic equation has a root in $\left] { - 1,1} \right[$     AG

(ii)     let $g(x) = 2{x^4} - 4{x^3} - 5{x^2} + 4x + 1$.

$g( - 1) = g(1) < 0$ and $g(0) > 0$     M1

as $g$ is a polynomial function it is continuous in $\left[ { - 1,0} \right]$ and $\left[ {0,{\text{ 1}}} \right]$.     R1

(or $g$ is a polynomial function continuous in any interval of real numbers)

then the graph of $g$ must cross the x-axis at least once in $\left] { - 1,0} \right[$     R1

and at least once in $\left] {0,1} \right[$.

[7 marks]