On what values of x does the graph of
f(x) = 2x^3 - 3x^2 + 12x + 87 have a horizontal tangent?
Take the derivative of the function because the derivative is equal to the slope of the tangent line.
a horizontal line has a slope of zero.
Set you derivative = 0 and solve for x. That should give you the correct value of x.
To find the values of x where the graph of a function has a horizontal tangent, we need to find the critical points of the function. A critical point occurs when the derivative of the function is equal to zero or undefined.
Let's find the derivative of the function f(x) first. Taking the derivative of each term using the power rule, we get:
f'(x) = 6x^2 - 6x + 12
Now, we need to find the values of x where f'(x) = 0. To do this, we set f'(x) equal to zero and solve for x:
6x^2 - 6x + 12 = 0
Next, we can simplify the equation by dividing both sides by 6:
x^2 - x + 2 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. However, in this case, the equation cannot be factored easily, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation, a = 1, b = -1, and c = 2, we can solve for x:
x = (-(-1) ± √((-1)^2 - 4(1)(2))) / (2(1))
x = (1 ± √(1 - 8)) / 2
x = (1 ± √(-7)) / 2
Since we have a negative value under the square root (square roots of negative numbers are imaginary), there are no real solutions. Therefore, the equation f'(x) = 0 has no solutions.
Now, we need to check for any values of x where the derivative is undefined. This occurs when there is a vertical tangent or a discontinuity in the function. However, in this case, our function is a polynomial, so it is defined for all real numbers, and we don't have any vertical tangents or discontinuities.
Therefore, the function f(x) = 2x^3 - 3x^2 + 12x + 87 does not have any values of x where the graph has a horizontal tangent.