Quick questions on Rational functions and asymptotes - TCE Mathematics Specialised (Tasmania)

1short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is locating turning points by calculus?

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For a rational function with an oblique asymptote, the cleanest route to turning points is to write the function in divided form

f(x) = (mx + c) + \dfrac{r}{x - a}

and differentiate. The TASC questions above all reduce to

f'(x) = 1 - \dfrac{r}{(x - a)^2} = 0

, giving

(x - a)^2 = r

and two symmetric critical points either side of the vertical asymptote. The branch above the asymptote carries a local minimum and the branch below carries a local maximum, which is a useful sanity check on the sign of the second derivative.

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