The tangent lines of the cubic function f(x) = x(x - 2)(x - 6) at the zeros 0, 2 and 6 are y = 12x, y = -8x + 16 and y = 24x - 144
Part A: The tangent lines of f(x) = x(x - 2)(x - 6)
The cubic function is given as:
f(x) = x(x - 2)(x - 6)
Expand
f(x) = x³ - 8x² + 12x
Differentiate the function
m = 3x² - 16x + 12
At the point x = 0, we have:
m = 3(0)² - 16(0) + 12
m = 12
Also, we have:
f(0) = 0
A linear equation is represented as:
y = m(x - x₁) + y₁
So, we have:
y = 12(x - 0) + 0
y = 12x
At the point x = 2, we have:
m = 3(2)² - 16(2) + 12
m = -8
Also, we have:
f(2) = 0
A linear equation is represented as:
y = m(x - x₁) + y₁
So, we have:
y = -8(x - 2) + 0
y = -8x + 16
At the point x = 6, we have:
m = 3(6)² - 16(6) + 12
m = 24
Also, we have:
f(6) = 0
A linear equation is represented as:
y = m(x - x₁) + y₁
So, we have:
y = 24(x - 6) + 0
y = 24x - 144
The above means that the tangent lines are:
y = 12x, y = -8x + 16 and y = 24x - 144
From the attached graph, we can see that the tangent lines at the zero values x = 0 and x = 2 intersect the graph of f(x) at the zero value x = 6
Part B: The tangent lines of f(x) = (x - a)(x - b)(x - c)
The equation is given as:
f(x) = (x - a)(x - b)(x - c)
To prove the statement using a computer algebra system, simply follow the next steps:
- Set values for a, b and c
- Include the following vertices in the plane: (a,f(a)), (b,f(b)) and (c,f(c))
- Differentiate f(x) i.e g(x) = f'(x)
- Include the following vertices in the plane: (a,g(a)), (b,g(b)) and (c,g(c))
- Lastly, plot the following equations: y = g(a)(x - a) + f(a), y = g(b)(x - b) + f(b) and y = g(c)(x - c) + f(c)
See attachment (2) for proof
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