The function f(x)= ax^3 - bx +c passes through the origin, f(-1)=4/3 and it has an extreme point at x=1
(i) Find the values of a, b and c.
(ii) Sketch the graph
(iii) Find the area bounded by the graph of f(x) and the x-axis between the lines x=-1 and x=1
To find the values of a, b, and c in the function f(x) = ax^3 - bx + c, we can use the given information and solve a system of equations.
(i) Since the function passes through the origin, we know that f(0) = 0. Plugging in x = 0 into the function, we get:
f(0) = a(0)^3 - b(0) + c
0 = 0 - 0 + c
c = 0
Now, we can rewrite the function as f(x) = ax^3 - bx. We still need to find the values of a and b.
Since the function has an extreme point at x = 1, its derivative at x = 1 must be equal to 0. Taking the derivative of f(x), we get:
f'(x) = 3ax^2 - b
Plugging in x = 1 and setting f'(1) = 0, we have:
0 = 3a(1)^2 - b
0 = 3a - b
Substituting c = 0 and the equation 0 = 3a - b into the equation f(-1) = 4/3, we can solve for a and b:
f(-1) = a(-1)^3 - b(-1) + c
4/3 = -a - b
4/3 = -a - (3a - b)
4/3 = -a - 3a + b
4/3 = -4a + b
Now, we have a system of two equations:
3a - b = 0 (from f'(1) = 0)
-4a + b = 4/3 (from f(-1) = 4/3)
To solve this system, we can use either substitution or elimination method. Let's use the elimination method:
Adding the two equations, we get:
3a - b + (-4a + b) = 0 + 4/3
-a = 4/3
a = -4/3
Substituting a = -4/3 into the first equation, we find:
3(-4/3) - b = 0
-4 - b = 0
b = -4
Therefore, the values of a, b, and c are:
a = -4/3
b = -4
c = 0
(ii) To sketch the graph of f(x) = ax^3 - bx + c, we can use the values we found. The graph passes through the origin and has an extreme point at x = 1. Plugging in the values, we have:
f(x) = (-4/3)x^3 - (-4)x + 0
f(x) = (-4/3)x^3 + 4x
The graph will be a cubic function that passes through the origin and has a point of inflection at x = 1.
(iii) To find the area bounded by the graph of f(x) and the x-axis between the lines x = -1 and x = 1, we can integrate the function over that interval and take the absolute value.
The area is given by:
Area = ∫[from -1 to 1] |f(x)| dx
Plugging in the function f(x) = (-4/3)x^3 + 4x, we have:
Area = ∫[from -1 to 1] |(-4/3)x^3 + 4x| dx
Now, we evaluate this integral to find the area.