A curve $\displaystyle y=x^2e^{-kx}$, where $\displaystyle k$ is a positive constant has stationary points of the curve are at (0,0) and (8,a).
Determine the exact values of the constants $\displaystyle k$ and $\displaystyle a$.
A curve $\displaystyle y=x^2e^{-kx}$, where $\displaystyle k$ is a positive constant has stationary points of the curve are at (0,0) and (8,a).
Determine the exact values of the constants $\displaystyle k$ and $\displaystyle a$.
Product rule? Just in case a picture helps...
Or, zooming in on the chain rule...
... where (key in spoiler) ...
Spoiler:
Spoiler:
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
From Tom's differentiation we have [tex]f'(x) = 2xe^{-kx}-kx^2e^{-kx} = 0 \Rightarrow e^{-kx}\left(2x-kx^2\right) = 0[/Math].
Since $\displaystyle e^{-kx} \ne 0$ for any real number $\displaystyle k$, we have $\displaystyle 2x-kx^2 = 0$. We know that $\displaystyle f'(8) = 0$,
so $\displaystyle 16-64k = 0 \Rightarrow k = \frac{1}{4}$. You can find $\displaystyle a$ easily now as it's just $\displaystyle f(8)$.