Consider the following function.
f(x) = 7x3 + x2 - 0.08x + 12
Find the two points on the graph at which the tangent line is horizontal. Give your values correct to 2 decimal places.
( , ) (point with smaller x value)
( , ) (point with larger x value)
thanks so much bob!
I suggest youtake the first derivative, which is slope, set that equal to zero, and solve for x.
To find the points on the graph where the tangent line is horizontal, we need to find the values of x where the derivative of the function is equal to zero.
Step 1: Calculate the derivative of the function, f'(x).
The derivative of the function f(x) = 7x^3 + x^2 - 0.08x + 12 can be found by taking the derivative of each term separately.
Using the power rule, the derivative of 7x^3 is 21x^2.
The derivative of x^2 is 2x.
The derivative of -0.08x is -0.08.
The derivative of a constant term (in this case, 12) is zero.
Therefore, the derivative of f(x) is f'(x) = 21x^2 + 2x - 0.08.
Step 2: Set the derivative equal to zero.
To find the values of x where the tangent line is horizontal, we set the derivative equal to zero and solve for x.
21x^2 + 2x - 0.08 = 0
Step 3: Solve the equation.
To solve this quadratic equation, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 21, b = 2, and c = -0.08. Plugging these values into the quadratic formula, we get:
x = (-2 ± √(2^2 - 4*21*(-0.08))) / (2*21)
x = (-2 ± √(4 + 6.72)) / 42
x = (-2 ± √10.72) / 42
Simplifying further:
x = (-2 ± √(2.72*4)) / 42
x = (-2 ± 2√(0.68)) / 42
x = (-1 ± √(0.68)) / 21
Now we have two possible values for x where the tangent line is horizontal.
Step 4: Calculate the corresponding y-values.
To find the corresponding y-values for the two points, we substitute the x-values we found into the original function f(x) = 7x^3 + x^2 - 0.08x + 12.
For the point with the smaller x-value (x1):
f(x1) = 7(x1)^3 + (x1)^2 - 0.08(x1) + 12
For the point with the larger x-value (x2):
f(x2) = 7(x2)^3 + (x2)^2 - 0.08(x2) + 12
Evaluating these expressions will give us the corresponding y-values for the two points.
Finally, round the x and y values to two decimal places for the final answer.