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

Calculus - Differentiation - Applications of Calculus.
2nd derivative - Concavity -Test Yourself 1.


 

On this page, the questions address:
1. Finding the concavity.
2. Finding points of concavity change.
3. Interpreting concavity.
4. Using 1st and 2nd derivative combinations.
5. Finding the maximum gradient.

 

Find the concavity. 1. (i) Find the concavity function for the curve given by y = x3(x - 2).

(ii) What is the concavity of this curve at the points where the curve crosses the x axis?

 2. Determine the value of the concavity for the function

y = (1 - 3x)3

at x = 1 and interpret its value.
Finding points of concavity change. 3. Find the point(s) on the curve

y = x3 - 3x2 - 9x + 11

where the concavity changes sign.

 4. Find the point(s) on the curve

y = 12x3 -3x4 + 11

where the concavity changes sign.
  5. Find the regions on the curve

y = x3 - x2 - 4x - 3 where the curve is

(i) concave up;

(ii) concave down.

 6. Find the regions on the curve
y = 27x - x3 where the the curve is

(i) concave up;

(ii) concave down.
  7. The curve y = 2x3 + ax2 - 3 has a point of inflexion at x = 1.
What is the value of a?
Answer. a = -6.		  
Interpreting concavity of a curve. 7. Show that the curve

is always concave down.

 8. Show that the curve

is always concave up when x > 0.
  9. For the curve y = 3 + 4x3 - x4:

(i) find the first two derivatives;

(ii) find the x values which solve the equations f '(x) = 0 and f "(x) = 0;

(iii) with reference to the nature of this function, comment on the shape of the curve at the three values you have found.

(iv) Sketch the function.

 

 10. For the curve
y = 2x4 - 4x3 + 2x2 + 10:

(i) find the first two derivatives;

(ii) find the x values which solve each of the equations f '(x) = 0 and
f "(x) = 0;

(iii) comment on the shape of the curve at the five x values you have found.
Using 1st and 2nd derivative combinations. 11. The first and second derivative functions of a curve are:

f '(x) = x(x - 3)(x + 5)

f "(x) = 3x2 + 4x -15

(i) Find the x values for which the gradient of the original curve is zero.

(ii) Find the x values for which the curve is concave down.

 12. If g(x) = x + 2x2, solve the equation

g '(a) = g "(a).
  13. Find the values for a, b and c if the curve
y = x3 + ax2 - bx + c has a
  • intercept at x = 1;
  • a stationary point at x = -2 and
  • a point of inflection at x = -0.5
 14. Given the equation y = x3 - x2 - x + 6,

find the x values where the curve is both decreasing and concave up.
  15. The graph of f(x) = 7 + 5x - x2 - x3 is defined in the domain [-3, 3].

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

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

(iii) Hence sketch the curve y = f(x) in the given domain.

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

Answer.(i) Maximum at (1, 10) and
minimum at (-5/3, 14/27).
(ii) POI at (-1/3, 148/27).
(iv) Concave up for (-3, -1/3).		 16. Given the curve :

(i) Find y' and y".

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

(iii) Sketch the curve.

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

Answer.(ii) Min at (2, 4)
Max at (-2, -4),
(iv) Concave up for x > 0.
  17. Consider the function
f(x) = x3 - x2 - 5x + 1.

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

(ii) Find any points of inflexion.

(iii) Sketch the curve y = f(x) for
[-2, 2] clearly indicating the endpoints. You do not need to find the x-intercepts.

(iv) For what values of x is the curve
y = f(x) decreasing but concave up?

Answer.(i) Max at (-1, 4)
Min at (5/3, -5.48).
(ii) POI at (1/3, 0.74).
(iv) [1/3, 2].		 18. For the function f(x) = 8x3 - 8x2:

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

(ii) Find the co-ordinates of any points of inflexion.

(iii) Sketch the graph of the function
y = f(x) showing stationary points, points of inflexion and x and y intercepts.

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

Answer.(i) Max at (0, 0)
Min at (2/3, -1.185).
(ii) POI at (1/3, -0.59).
(iv) 0 < x < 1/3.
Finding the maximum gradient. 19. Find the maximum gradient of the curve
y = x3 - 6x2 + 5.		 20. The amount M of medication of a certain medication present in a person's blood after t hours is described by the equation

M = 9t2 - t3 for 0 ≤ t ≤ 9

When is the amount of medicine in the blood increasing most rapidly?

Answer.t = 3.
  21. Prove that the graph of

y = ax3 + bx2 + cx + d
has two distinct turning points if b2 > 3ac.

Find the values of a, b, c and d for which the graph of this form has turning points at (0.5, 1) and at (1.5, -1)

22.