Brief Calculus.

Question

Use Calculus to show the x-values where the tangent line is horizontal

f(x) = x^2(4-x^2)

Answer 1

Horizontal lines have a slope = 0, and since the first derivative provides us a function that gives the slopes of the tangent lines to the curve, we can take the first derivative, using power rule, set it = 0, and solve.

Note two things: Though we could take the first derivative using product rule, it is easier to simply distribute the x² in the given function rule before proceeding. Also, this quartic (degree 4) polynomial will have at most 3 horizontal tangents (i.e. 3 turning points at most.

Actually, another feature of the given function worth noting is that it is even, which means it has y-axis symmetry. So it may have at most two non-zero turning points, as well as a turning point on the y-axis (x = 0).

f(x) = - x⁴ + 4x²

f^'(x) = -4x³ + 8x = 0

-4x(x² - 2) = 0

x = - √2 , 0 , and √2