A curve has equation $$$y=\mathrm{f}(x)$$$ and it is given that $$\[ \mathrm{f}^{\prime}(x)=\left(\frac{1}{2} x+k\right)^{-2}-(1+k)^{-2}, \]$$ where $$$k$$$ is a constant. The curve has a minimum point at $$$x=2$$$. It is now given that $$$k=-3$$$ and the minimum point is at $$$\left(2,3 \frac{1}{2}\right)$$$. Find $$$\mathrm{f}(x)$$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w21_qp_13 Year：2021 Question No：10(b)

Answer:

$\left[f(x)=\int\left(\left(\frac{1}{2} x-3\right)^{-2}-(-2)^{-2}\right) d x=\right]\left\{\frac{\left(\frac{1}{2} x-3\right)^{-1}}{-1 \times \frac{1}{2}}\right\}\left\{-\frac{x}{4}\right\}$
$3 \frac{1}{2}=1-\frac{1}{2}+c$
$f(x)=\frac{-2}{\left(\frac{1}{2} x-3\right)}-\frac{x}{4}+3$

Knowledge points:

1.8.1 understand integration as the reverse process of differentiation, and integrate (for any rational n except-1 , together with constant multiples, sums and differences

Answer:

Knowledge points:

Solution:

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