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Exam-Style Question on Calculus

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Question id: 569. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is allowed.

Consider the function \(f\) defined by \(f(x) = 25e^{x-5}\) for \(x \in \mathbb{R}^+\).

(a) Find the coordinates of the points where the graph of \(f\) intersects the line \(y=x\).

The line \(L\) has a gradient of \(-1\) and is a normal to the graph of \(f\) at the point \(R\).

(b) Find the exact coordinates of \(R\).

(c) Show that the equation of the line \(L\) is \(y=-x+6- \ln{25}\).

(d) Find the area of the region enclosed by the graph of \(f\) and its inverse.

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