Calculus  Differentiation  Applications of Calculus.
Given an equation  Test Yourself 2.
The questions of this page focus on the following issues: 
1. Finding stationary points. 
2. Where is a curve increasing/decreasing. 
3. The concavity of the curve. 
4. The points of inflexion. 
5. The number of solutions for an equation. 
Stationary points and sketching.  1. A function is defined as f(x) = x^{3}  3x^{2}.
Answer. (i) Maximum at (0,0); 
2. For the cubic function y = x^{3}  x^{2}  x + 6


3. The function f(x) = 2x^{3} + 3x^{2}  12x + 7 is defined with the domain [3. 3].
Answer. (i) Maximum at (2,27); minimum at (1, 0). (ii) POI at (0.5, 13.5). (iv) Max value in domain is 52. (v) 0.5 < x < 1. 

4. The function f(x) = x^{3}  6x^{2} + 9x 4 is defined in the domain [0, 4].


.  5. Given that f(x) = (x  2)^{2}(3  x):
Answer. (ii) Maximum at (8/3,4/27); minimum at (2, 0). (iIi) Intercepts at (2, 0) and at (0, 12). 
Absolute maximum & minimum values  
Curve increasing/decreasing.  6. (i) For the curve y = 3x  x^{3}, find the stationary points and determine their nature.
Answer. (i) Maximum at (1,2); 
Concavity  7. A function is defined by f(x) = x^{3}  3x^{2}  9x  22.
Answer. (i) TPs at (1, 27) max and (3, 5) min; (ii) POI at (1, 11). (iv) x > 1. 
8. For what values of x is the curve f(x) = 2x^{3} + x^{2} concave down?
Answer. Concave down for x < 1/6. 

Points of inflexion.  9. The diagram below shows a sketch of the curve y = 6x^{2}  x^{3}.
The curve cuts the xaxis at C. It also has point of inflexion at A and a local maximum at B.

10.  
Number of solutions.  11. 
12. 