Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Applications of Calculus.
Max/min of given equations - Test Yourself 1.


 

The questions on this page require the determination of both the first and second derivative of given equations.

The questions of this page focus on the following issues:
1. Finding a minimum & maximum values.
2. Absolute maximum/minimum values.
3. Where is a curve increasing/decreasing.
4. The number of solutions for an equation.

 

Minimum & maximum values. 1. The function y = x3 - 3x2 - 9x + 1 is defined in the domain [-2, 5].

(i) Find the coordinates of any turning points and determine their nature.

(ii) Find the coordinates of any points of inflexion.

(iii) Draw a neat sketch of the curve.

Answer.(i) Maximum at (-1, 6) and
minimum at (3, -26).
(ii) POI at (1, -10). 		2. The cost of running a tour ($C) for x people can be expressed as C = x3 - 972x + 20,000.

(i) Find the number of people required to make the tour cost as low as possible.

(ii) Find the cost per person when the tour is of minimum cost.

Answer.(i) Need 18 people for min cost.
(ii) Min cost/person = $463.11.
  3. A curve is determined by the equation

y = x3 - 12x + 4.

(i) Find the coordinates of any stationary points and determine their nature.

(ii) Find the coordinates of any points of inflexion.

(iii) Sketch the curve for the domain [-3, 4].

Answer.(i) Maximum at (-2, 20) and
minimum at (2, -12).
(ii) POI at (0, 24).
(iv) Maximum value of 20
at x = -2 and at x = 4.		 4. Consider the curve y = 20 - 5x2 - x3
for [-5, 2].

(i) Find the stationary points and determine their nature.

(ii) Find the point of inflection.

(iii) Sketch the curve for [-5, 2].

Answer.(i) Maximum at (0, 20) and
minimum at (-10/3, 1.48).
(ii) POI at (-5/3, 10.74).
(iv) Maximum value is 20 at both
x = -5 and at x = 0.
Absolute max/mins. 5. For the function f(x) = 3x2 - x3 + 9x - 2:

(i) Find f '(x) and f"(x).

(ii) Find the coordinates of the stationary points and determine their nature.

(iii) Sketch the graph of the function in the domain [-2, 5].

(iv) What is the minimum value for the function in this domain?

Answer.(ii) Maximum value at (3, 25)
Minimum value at (-1, -7).
(iv) Minimum value of the function is -7
and that minimum occurs
at x = -1 and at x = 5
in the given domain.		 6.
  7. Consider the curve f(x) = x3 - 2x2 + x + 4 for [-2, 2].

(i) Find the stationary points and determine their nature.

(ii) Find the points of inflection.

(iii) Find the points of intersection with each axis.

(iv) Sketch the curve for [-2, 2].

(v) What is the maximum value for f(x) in the interval [-2, 2].

Answer.(i) Maximum at (1/3, 112/27) and
minimum at (1, 4).
(ii) POI at (2/3, 110/27).
(iii) (0, 4) and (-1, 0)
(v) Max value is f(x) = 6.		 8. The function y = 2x3 + 3x2 - 36x + 5 is defined in the domain [-5, 5].

(i) Find the coordinates of any turning points and determine their nature.

(ii) Find the coordinates of any points of inflexion.

(iii) Draw a neat sketch of the curve.

(iv) Determine the maximum value of the function in the given domain.

Answer.(i) Maximum at (-3, 86) and
minimum at (2, -39).
(ii) POI at (-0.5, 23.5).
(iv) Maximum value is 150.
 

9. Find both the relative and the absolute maximum and minimum values for the function

f(x) = x3 - 6x2 + 9x - 5 = 0

in the domain [0, 4].

 10. Find both the relative and the absolute maximum and minimum values for the function

f(x) = 3x4 + 4x3 -12x2 - 3

in the domain [-2, 2].

Answer.The local minimum and absolute minumum are -35 at x = -2.
The local maximum is -8 when
x = 0 and the absolute maximum in the domain is 29.
  11. Find the absolute maximum and minimum values of the function in the domain [1, 9].
Answer.The minimum value is 1 at x = 1 and the maximum value is 5
at x = 9.
We cant do calculus with this question to obtain the values!!!.		 12. Find the absolute maximum and minimum values of the function in the domain [1, 9].
Answer.The minimum value is 2/3 at x = 1 and the maximum value is 2/19
at x = 9.
We can't do calculus with this question to obtain the values!!!.
Curve sketching. 13. A function is defined by .

(i) Find the coordinates of the stationary points of the graph y = f(x) and determine their nature.

(ii) Sketch the graph of y = f(x) showing all its essential features including stationary points and intercepts.

(iii) For what values in the domain is the curve decreasing?

Answer.(ii) Maximum at (0, 0) and
minimum at (4, -6.4).
(iii) curve is decreasing 0 ≤ x ≤ 4.		 14. For the curve :

(i) Find the stationary points and determine their nature.

(ii) Sketch the curve in the domain
showing all relevant information.

(iii) For what values of x is the curve rising?
  15. Let f(x) = x3 + kx2 + 3x - 5 where k is a constant.

Find the values of k for which f(x) has NO stationary points.

Answer.-3 < k < 3.		 16. (i) Find the derivative of

(ii) Find the value(s) of m for which the graph of

has no stationary points.

Answer.-5 < m < 3.
  17. For the curve y = 7 + 5x - x2 - x3,

(i) find the co-ordinates of the maximum and minimum values.

(ii) Find the point where the concavity changes.

(iii) Sketch the curve.

(iv) For what values of x is the curve concave down?

 18. For the curve
(i) Find the co-ordinates of where the maximum and minimum values occur and determine their nature.

(ii) Sketch the curve.

(iii) For what x values is the curve concave up.
  19. For the curve y = 3x4 - 8x3 + 6:

(i) Find the turning points of the curve and determine their nature.

(ii) Find the points of inflection of the curve.

(iii) Sketch the curve showing the turning points and the points of inflection (do not find the x-intercepts).

Answer.(i) Min at (2, -10)
(ii) HPOI at (0, 6).
and POI at (4/3, -3.48).		 20. Given f(x) = (x + 2)(x - 2)3

and f '(x) = 4(x - 2)2(x + 1) = 4x3 - 12x2 + 16

(i) Find the stationary points of f(x) and determine their nature.

(ii) Find the coordinates of any points of inflection.

(iii) Sketch the graph of y = f(x) indicating the information you have obtained above.

(iv) Sketch the graph of the derivative on a separate set of axes showing the correspondences for the x values identified previously.

Answer.(i) Min at (-1, -27)
(ii) HPOI at (2, 0)
POI at (0, -16).
Number of solutions. 21. Consider the curve y = 2x2 (x - 3)2.

(i) Find any stationary points and determine their nature.

(ii) Sketch the curve for -2 ≤ x ≤ 4.

(iii) Determine the number of solutions for the equation y = 2x2 (x - 3)2 + x - 2 = 0.

Answer.(i) Max at (1.5, 10.125)
Min at (0, 0) and
at (3, 0).
(iii) No. of solutions = 2.		 22. A function can be expressed as

f(x) = 3x4 + 4x3 - 12x2

(i) Find the coordinates of any stationary points of the function and determine their nature.

(ii) Sketch the graph of f(x) showing the stationary points.

(iii) For what values of x is the function increasing?

(iv) For what values of k will the equation

f(x) = 3x4 + 4x3 - 12x2 + k = 0 have no solution?

(v) For what values of k will the equation
f(x) = 3x4 + 4x3 - 12x2 + k = 0 have exactly three solutions?

Answer.(i) Max at (0, 0)
Min at (-2, 32) and at (-1, 5).
(iii) Increasing for -2 < x < 0
and for x > 1.
(iv) k < -32.(v) k = 0 or k = -5.