For what values of x is the tangent line of the graph of
f(x)=8x3+36x2+46x−72
parallel to the line y=−2x+1.9 ? Enter the x values in order, smallest first, to 4 places of accuracy:
x1= ≤ x2=
f(x) = 8x³ + 36x² + 46x - 72
f'(x) = 24x² + 72x + 46
The slope of y = -2x+1.9 is -2
So, we want to find values of x where f'(x) = -2
24x² + 72x + 46 = -2
24x^2 + 72x + 48 = 0
x^2 + 3x + 2 = 0
(x+1)(x+2)
So, x = -2.0000, -1.0000
To find the values of x for which the tangent line of the graph of f(x) is parallel to the line y = -2x + 1.9, we need to determine the derivative of f(x) and find the values of x where the derivative is equal to -2.
1. Find the derivative of f(x):
f'(x) = d/dx (8x^3 + 36x^2 + 46x - 72)
= 24x^2 + 72x + 46
2. Set the derivative equal to -2:
24x^2 + 72x + 46 = -2
3. Rearrange the equation:
24x^2 + 72x + 48 = 0
4. Solve for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
= (-72 ± √(72^2 - 4 * 24 * 48)) / (2 * 24)
Calculating the values of x using the quadratic formula, we get:
x1 ≈ -1.9996
x2 ≈ -0.0004
Therefore, the values of x for which the tangent line of the graph of f(x) is parallel to y = -2x + 1.9 are approximately x1 ≈ -1.9996 and x2 ≈ -0.0004.
To find the values of x for which the tangent line of the graph of f(x) is parallel to the line y = -2x + 1.9, we need to find the slope of the tangent line of f(x) and compare it to the slope of the given line.
The slope of the given line y = -2x + 1.9 is -2. We want the tangent line of f(x) to have the same slope.
The slope of the tangent line of a function f(x) at a specific point is given by the derivative of f(x) evaluated at that point.
Step 1: Find the derivative of f(x).
f(x) = 8x^3 + 36x^2 + 46x - 72
To find the derivative, we can take the derivative of each term separately using the power rule:
f'(x) = 3(8)x^(3-1) + 2(36)x^(2-1) + 1(46)x^(1-1) - 0
f'(x) = 24x^2 + 72x + 46
Step 2: Set the derivative equal to the slope we want, which is -2.
24x^2 + 72x + 46 = -2
Step 3: Rearrange the equation and solve for x.
24x^2 + 72x + 48 = 0
We can simplify the equation by dividing both sides by 24, resulting in:
x^2 + 3x + 2 = 0
We can factor this quadratic equation:
(x + 1)(x + 2) = 0
By setting each factor equal to zero, we have two possible solutions:
x + 1 = 0 --> x = -1
x + 2 = 0 --> x = -2
Therefore, the values of x for which the tangent line of the graph of f(x) is parallel to the line y = -2x + 1.9 are x = -1 and x = -2.
So, the values of x in order, from smallest to largest, are:
x1 = -2
x2 = -1