| Marks available | 8 |
| Reference code | 14M.3ca.hl.TZ0.4 |
Question
The function f is defined by \(f(x) = \left\{ \begin{array}{r}{e^{ – x^3}}( – {x^3} + 2{x^2} + x),x \le 1\\ax + b,x > 1\end{array} \right.\), where \(a\) and \(b\) are constants.
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 \(\left] { – 1,1} \right[\).
Markscheme
\(\mathop {{\text{lim}}}\limits_{x \to {1^ – }} {{\text{e}}^{ – {x^2}}}\left( { – {x^3} + 2{x^2} + x} \right) = \mathop {{\text{lim}}}\limits_{x \to {1^ + }} (ax + b)\) \(( = a + b)\) 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]
Examiners report
| Marks available | 6 |
| Reference code | 18M.3ca.hl.TZ0.2 |
Question
The function \(f\) is defined by
\[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{\left| {x – 2} \right| + 1}&{x < 2} \\
{a{x^2} + bx}&{x \geqslant 2}
\end{array}} \right.\]
where \(a\) and \(b\) are real constants
Given that both \(f\) and its derivative are continuous at \(x = 2\), find the value of \(a\) and the value of \(b\).
Markscheme
considering continuity at \(x = 2\)
\(\mathop {{\text{lim}}}\limits_{x \to {2^ – }} f\left( x \right) = 1\) and \(\mathop {{\text{lim}}}\limits_{x \to {2^ + }} f\left( x \right) = 4a + 2b\) (M1)
\(4a + 2b = 1\) A1
considering differentiability at \(x = 2\)
\(f’\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{ – 1}&{x < 2} \\
{2ax + b}&{x \geqslant 2}
\end{array}} \right.\) (M1)
\(\mathop {{\text{lim}}}\limits_{x \to {2^ – }} f’\left( x \right) = – 1\) and \(\mathop {{\text{lim}}}\limits_{x \to {2^ + }} f’\left( x \right) = 4a + b\) (M1)
Note: The above M1 is for attempting to find the left and right limit of their derived piecewise function at \(x = 2\).
\(4a + b = – 1\) A1
\(a = – \frac{3}{4}\) and \(b = 2\) A1
[6 marks]