June 2023 Paper 1 Q6

Notes

M1: Attempt to differentiate using the chain rule
Obtain derivative of form \(\mathrm{f}(x)\mathrm{e}^{x^2 + 3x}\)
Could also split into two terms and use product rule to obtain derivative of form \(\mathrm{f}(x)\mathrm{e}^{x^2}\mathrm{e}^{3x} + k\mathrm{e}^{x^2}\mathrm{e}^{3x}\ (k \neq 0)\)
M0 if attempt to split results in sum not product

A1: Obtain correct derivative
Brackets must be seen, or implied by later work
aef eg \(\left(2x\mathrm{e}^{x^2}\right)\left(\mathrm{e}^{3x}\right) + \left(\mathrm{e}^{x^2}\right)\left(3\mathrm{e}^{3x}\right)\) from splitting into two terms first
Could be in terms of \(u\), as long as \(u\) clearly defined

A1: Equate correct derivative to 0 soi and solve to obtain \(x = -\frac{3}{2}\)
ISW any \(y\)-coordinates if given
A0 if any additional solutions for \(x\)
Must see differentiation, so \(x = -\frac{3}{2}\) with no supporting method gets no credit (as question is ‘determine’)

B1FT: Indicate no solutions from the exponential term
FT their derivative as long as of form \(\mathrm{f}(x)\mathrm{e}^{x^2 + 3x}\) or \(\mathrm{f}(x)\mathrm{e}^u\)
Allow BOD for explanations such as \(\mathrm{e}^x \gt 0\) for all \(x\)
Must have some reason, eg ‘\(\mathrm{e}^{x^2 + 3x}\) is always positive’, ‘\(\mathrm{e}^{x^2 + 3x}\) cannot be negative’, ‘cannot take ln of a negative number’, ‘not defined’, ‘not real’, ‘no solutions’
A0 for ‘math error’ or ‘doesn’t work’

Alternative method

M1: Take ln and attempt implicit differentiation
Must deal correctly with \(\ln y\)

A1: Obtain correct derivative
May still have \(\dfrac{1}{y}\) on LHS

A1: Equate correct derivative to 0 soi and solve to obtain \(x = -\frac{3}{2}\)

B1: Indicate no solutions from the exponential term
See main MS for guidance
Could also explain why no solutions from \(\dfrac{1}{y}\)

Notes

M1: Attempt to differentiate again using the product rule correctly
Obtain derivative of form \(\left(ax^2 + bx + c\right)\mathrm{e}^{x^2 + 3x}\) aef

A1: Obtain correct derivative
aef eg (depending on method) \(2\mathrm{e}^{x^2 + 3x} + 2x(2x + 3)\mathrm{e}^{x^2 + 3x} + 3(2x + 3)\mathrm{e}^{x^2 + 3x}\)
or \(\left(\left(2\mathrm{e}^{x^2} + 4x^2\mathrm{e}^{x^2}\right)\mathrm{e}^{3x} + \left(2x\mathrm{e}^{x^2}\right)3\mathrm{e}^{3x}\right) + \left(6x\mathrm{e}^{x^2}\mathrm{e}^{3x} + 9\mathrm{e}^{x^2}\mathrm{e}^{3x}\right)\)
Could be in terms of \(u\), as long as \(u\) clearly defined

B1: State, or clearly imply, correct condition at any point in proof
Must be general statement, and not just \(\gt 0\) from testing the stationary point

M1: Explain why correct quadratic is always positive
Could note minimum value of 2 as completed square form
Could use expanded quadratic, which should be \(4x^2 + 12x + 11\)
If showing no real roots then must also say that it is a positive quadratic
Condone \(\gt\) / \(\geqslant\) muddles for M1 only
Could show that there are no points of inflection and \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} \gt 0\) for at least one point

A1: Full and convincing proof to show that curve is convex for all \(x\)
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Alt method for first 2 marks

M1: Attempt second derivative, using implicit differentiation and the product rule
If still \(\dfrac{1}{y}\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2x + 3\) then must be a correct attempt to differentiate the LHS

A1: Obtain correct derivative
aef
Will need to use \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = y(2x + 3)\) to make further progress