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Question id: 208. This question is similar to one that appeared in an IB Studies paper in 2015. The use of a calculator is allowed.

### Differentiation

Consider the function $$f(x)=\frac{20}{x^2}+kx$$ where $$k$$ is a constant and $$x\neq0$$.

(a) Write down $$f'(x)$$

The graph of $$y = f(x)$$ has a local minimum point at $$x=2$$.

(b) Show that $$k=5$$.

(c) Find $$f(1)$$.

(d) Find $$f'(1)$$.

(e) Find the equation of the normal to the graph of $$y=f(x)$$ at the point where $$x=1$$
Give your answer in the form $$ay+bx+c=0$$ where $$a, b, d \in \mathbb{Z}$$

(f) Sketch the graph of $$y=f(x)$$ , for $$-5\le x\le 10$$ and $$-10\le y\le 50$$.

(g) Write down the coordinates of the point where the graph of $$y=f(x)$$ intersects the x-axis.

(h) State the values of $$x$$ for which $$f(x)$$ is decreasing.

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