## Exam-Style Question on Differentiation## A mathematics exam-style question with a worked solution that can be revealed gradually |

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

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 \(a, b, c \in \mathbb{Z}\)

(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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