Final answer:
To find the points on the curve where the tangent line is horizontal, we need to find the critical points or points where the derivative of the function is equal to zero.
Step-by-step explanation:
To find the points on the curve where the tangent line is horizontal, we need to find the critical points or points where the derivative of the function is equal to zero. So, let's find the derivative of the function first.
y = x⁴ - 14x² + 3
dy/dx = 4x³ - 28x
Now, set the derivative equal to zero and solve for x:
4x³ - 28x = 0
Factor out x:
x(4x² - 28) = 0
Solve for x:
x = 0 or x² - 7 = 0
From the equation x² - 7 = 0, we get:
x = √7 or x = -√7
So, the points on the curve where the tangent line is horizontal are (√7, 7) and (-√7, 7).