2024 ONE(d)

Merit
Question
Find the xx-value(s) of any stationary points on the graph of the function below, and determine
their nature.

y=(2x−1)e−2xy = (2x - 1)e^{-2x}

You must use calculus and show any derivatives that you need to find when solving this problem.
Official Answer
dydx=e−2x(4−4x)\dfrac{dy}{dx}=e^{-2x}(4-4x)

d2ydx2=e−2x(8x−12)\dfrac{d^2y}{dx^2}=e^{-2x}(8x-12)

Solving dydx=e−2x(4−4x)=0\dfrac{dy}{dx}=e^{-2x}(4-4x)=0 gives x=1x=1

Then x=1x=1 gives d2ydx2=e−2<0\dfrac{d^2y}{dx^2}=e^{-2}<0

i.e. Maximum when x=1x=1.
Grading Criteria

Achievement (u)

  • Correct expression for dydx\dfrac{dy}{dx}.

Merit (r)

  • Finds x=1x=1 and justifies that this is a maximum using a calculus method, and with evidence of derivative.

Excellence T1

-

Excellence T2

-
Video Explanation
2024 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 7:38)
Subscribe
Exam Paper (2024)Assessment Schedule (2024)
Related Questions
2024 THREE(e)Excellence

The diagram below shows part of the graph of the function $f(x) = e^{-x^2}$, where $x \geq 0$. [DI...

2024 THREE(c)Merit

Find the $x$-value(s) of any stationary point(s) on the graph of the function $f(x)=\dfrac{x^2-5x+4...

2024 THREE(d)Merit

Jamie is doing some baking and pouring the flour to form a conical pile. The height o...

2024 TWO(e)Excellence

The graph of the function $y = \dfrac{xe^{3x}}{2x + k}$, where $k$ is a non-zero constant, has a sin...

2024 TWO(c)Merit

Show that $y = \sin(x^2) - \cos(x)$ is a solution to the equation $\dfrac{d^2y}{dx^2} + 4x^2\,y = 2...