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Question
Let \(f(x) = a{x^3} + b{x^2} + c\) , where a , b and c are real numbers. The graph of f passes through the point (2, 9) .
Show that \(8a + 4b + c = 9\) .
The graph of f has a local minimum at \((1{\text{, }}4)\) .
Find two other equations in a , b and c , giving your answers in a similar form to part (a).
Find the value of a , of b and of c .
Answer/Explanation
Markscheme
attempt to substitute coordinates in f (M1)
e.g. \(f(2) = 9\)
correct substitution A1
e.g. \(a \times {2^3} + b \times {2^2} + c = 9\)
\(8a + 4b + c = 9\) AG N0
[2 marks]
recognizing that \((1{\text{, }}4)\) is on the graph of f (M1)
e.g. \(f(1) = 4\)
correct equation A1
e.g. \(a + b + c = 4\)
recognizing that \(f’ = 0\) at minimum (seen anywhere) (M1)
e.g. \(f'(1) = 0\)
\(f'(x) = 3a{x^2} + 2bx\) (seen anywhere) A1A1
correct substitution into derivative (A1)
e.g. \(3a \times {1^2} + 2b \times 1 = 0\)
correct simplified equation A1
e.g. \(3a + 2b = 0\)
[7 marks]
valid method for solving system of equations (M1)
e.g. inverse of a matrix, substitution
\(a = 2\) , \(b = – 3\) , \(c = 5\) A1A1A1 N4
[4 marks]
Question
Let f(x) = ln x − 5x , for x > 0 .
Find f ’(x).
Find f ”(x).
Solve f ’(x) = f ”(x).
Answer/Explanation
Markscheme
\(f’\left( x \right) = \frac{1}{x} – 5\) A1A1 N2
[2 marks]
f ”(x) = −x−2 A1 N1
[1 mark]
METHOD 1 (using GDC)
valid approach (M1)
eg 
0.558257
x = 0.558 A1 N2
Note: Do not award A1 if additional answers given.
METHOD 2 (analytical)
attempt to solve their equation f '(x) = f ”(x) (do not accept \(\frac{1}{x} – 5 = – \frac{1}{{{x^2}}}\)) (M1)
eg \(5{x^2} – x – 1 = 0,\,\,\frac{{1 \pm \sqrt {21} }}{{10}},\,\,\frac{1}{x} = \frac{{ – 1 \pm \sqrt {21} }}{2},\,\, – 0.358\)
0.558257
x = 0.558 A1 N2
Note: Do not award A1 if additional answers given.
[2 marks]