Use the Second Derivative to determine if y=arcsec(6x^2) is concaved up, concaved down, or has no concavity at x=4.

Question

Use the Second Derivative to determine if y=arcsec(6x^2) is concaved up, concaved down, or has no concavity at x=4.

Answer 1

First find dy/dx then find the d²y/dx² then plug 4 into the 2nd derivative. If the value of the 2nd derivative at 4 is positive then it is concave up, if negative it is concave down if undefined there is no concavity. There is, in fact, a fourth possibility which is that the concavity exists, but is zero....

y = arcsec (6x²)

dy/dx = d/dx(6x²) / [ |(6x²)| ( (6x²)² - 1 )^1/2] = 12x / [ |6x²| ( 36x⁴ - 1 )^1/2] =2x / [ x² ( 36x⁴ - 1 )^1/2] = 2 / [ x ( 36x⁴ - 1 )^1/2] = 2 x^-1 ( 36x⁴ - 1 )^-1/2

d²y/dx² = d/dx (2 x^-1 ( 36x⁴ - 1 )^-1/2) = 2 x^-1 • d/dx [ ( 36x⁴ - 1 )^-1/2 ] + d/dx (2 x^-1)• [ ( 36x⁴ - 1 )^-1/2 ]

= 2 x^-1 • (-1/2) ( 36x⁴ - 1 )^-3/2 • d/dx (36x⁴ - 1)+ -1 • 2 x^-2 • [ ( 36x⁴ - 1 )^-1/2 ]

= 2 x^-1 • (-1/2) ( 36x⁴ - 1 )^-3/2 • 144x³ + -1 • 2 x^-2 • [ ( 36x⁴ - 1 )^-1/2 ]

... since the goal is to find the value at x=4 simplifying the 2nd derivative appears to be pointless, but if your class requires it then I leave that up to you...

if you plug 4 in for x you get approximately -0.004

Therefore the curve is concave down at x=4