Applications of Differentiation


21 4. Let y = f (x) = 3 − 5x − 2 x 2 , x ∈R.


(a) Find f ʹ(x) and hence, the co-ordinates of the local maximum of the curve y = f (x).


(b) Solve f (x) = 0.


(c) Plot y = 3 − 5x − 2x 2 and 2x + y = 0 on the same diagram.


(d) Use your graphs to estimate the solutions of 2 x 2 + 3x − 3 = 0, correct to one decimal place.


5. (a) Find the co-ordinates of the local maximum and local minimum of y = x 3 − 6 x 2 , x ∈R.


(b) Find where the curve crosses the axes.


(c) Plot the curve in the domain −2 ≤ x ≤ 6, x ∈R.


(d) Find the range of values for which the curve is decreasing.


6. Let y = f (x) = x 3 − 3 x 2 , x ∈ R.


(a) Find f ʹ(x) and hence, fi nd the co-ordinates of the local maximum and local minimum of the curve.


(b) Plot the graph of y = f (x) in the domain −1 ≤ x ≤ 3, x ∈R.


(c) Use your graph to:


(i) estimate the solutions of f (x) + 2 = 0, correct to one decimal place,


(ii) fi nd the range of values of x for which f ʹ(x) < 0.


7. Let f (x) = x 3 − 3 x 2 + 1, x ∈ R. (a) Find f (–1) and f (3). (b) Find f ʹ(x).


(c) Find the co-ordinates of the local maximum and local minimum of y = f (x).


(d) Plot the graph of y = f (x) in the domain −1 ≤ x ≤ 3, x ∈R.


(e) Use your graph to:


(i) estimate the range of values of x for which f (x) < 0, x > 0, giving your answers correct to one decimal place,


(ii) fi nd the range of values of x for which f ʹ(x) < 0.


8. Let y = f (x) = 2 x 3 − 5 x 2 − 4x + 3, x ∈ R. (a) Copy and complete the table:


x –1·5 –1 023 3·5 y


–9 (b) Find


___ dx .


dy


(c) Find the co-ordinates of the local maximum and local minimum of y = f (x).


(d) Plot y = f (x) in the domain −1·5 ≤ x ≤ 3·5, x ∈R.


(e) On the same diagram, draw the line with the equation x − y = 0.


(f) Use both graphs to estimate the solutions of 2 x 3 − 5 x 2 − 5x + 3 = 0, correct to one decimal place.


9. Write y = 2 x − 1 in the form y = k 2 x , where k is a constant, 0 < k < 1. Plot the graph of


y = 2 x − 1 in the domain −2 ≤ x ≤ 3, x ∈R. Use your graph to: (a) show that there are no solutions of 2 x − 1 = 0,


(b) estimate the solution of 2 x − 1 = 3, correct to one decimal place.


10. (a) For f (x) = x 2 , fi nd the local minimum point and plot the function in the domain −3 ≤ x ≤ 3, x ∈R.


(b) On the same diagram as y = f (x), plot g (x) = 2 x in the domain −3 ≤ x ≤ 3, x ∈R.


(c) Use your diagram to estimate the solutions of x 2 = 2 x , correct to one decimal place.


13·5


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