Final answer:
To find the points where the tangent is horizontal, differentiate the equation of the curve and set the derivative equal to zero. Solve for x to find the x-coordinates of the critical points. These x-coordinates correspond to the points on the curve where the tangent is horizontal.
Step-by-step explanation:
To find the points on the curve where the tangent is horizontal, we need to determine where the slope of the curve is equal to zero. In other words, we need to find the x-coordinates of the critical points.
To do this, we can differentiate the equation of the curve with respect to x and set the derivative equal to zero. By solving this equation, we can find the x-coordinates of the critical points. These x-coordinates correspond to the points on the curve where the tangent is horizontal.
For the given equation, ҁ = 8x^3 + 3x^2 - 4x + 4, we can differentiate it to find ҁ'(x) = 24x^2 + 6x - 4. Setting this equal to zero and solving for x, we get x = -0.667 and x = 0.25.
Therefore, the points on the curve where the tangent is horizontal are (-0.667, ҁ(-0.667)) and (0.25, ҁ(0.25)).