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Differentiation - Increasing and Decreasing Functions

Differentiation - Increasing and Decreasing Functions

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IB AA SL: core and optional content

Core: derivative-sign analysis directly from the original rational function. Optional: partial fractions.

  1. Express \(\frac{7x^{3} + x^{2} + 5x + 3}{(x^{2} + 3)(x^{2} - 1)}\) in partial fractions.
  2. Show that when \(x < -\sqrt{3}, \frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{7x^{3} + x^{2} + 5x + 3}{(x^{2} + 3)(x^{2} - 1)} \right) < 0\).

Video Solution:

Default solution

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Answer:(i) \(\displaystyle \frac{4x}{x^2+3}+\frac{2}{x-1}+\frac{1}{x+1}\) (ii) \(f'(x)<0\) for \(x<-\sqrt3\).

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