Pearson Edexcel IAL Mathematics P2.7.1 Applications of differentiation

Practise using first and second derivatives to analyse curves, optimisation models, stationary points, and intervals of increasing or decreasing behaviour.

Exam points

differentiate functions, then set dy/dx=0 to find stationary points or model values
use d²y/dx² or derivative sign changes to justify maxima and minima
solve f'(x)>0 or f'(x)<0 to give intervals of increase or decrease

[Maximum number: 3]

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Figure 3

Figure 3 shows
- the curve $C_{1}$ with equation $y=x^{3}-5 x^{2}+3 x+14$
- the circle $C_{2}$ with centre T

The point T is the minimum turning point of $C_{1}$
Using Figure 3 and calculus,

find the coordinates of T

The curve $C_{1}$ intersects the circle $C_{2}$ at the point A with x coordinate 2

$y=x^{3}-5 x^{2}+3 x+14 \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}=3 x^{2}-10 x+3=0$

M1

Roots are $3, \frac{1}{3} \Rightarrow$ when $x=3, y=" 3^{\prime \prime 3}-5 \times " 3^{\prime \prime 2}+3 \times " 3 "+14=\ldots$ .

dM1

Centre is (3,5)

A1

(3)

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P2.7.1 - Applications of differentiation question 1