Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.6 Exercises - Page 373: 72

$$\eqalign{ & {\text{Relative minimum }}\left( {0.6812, - 0.4466} \right) \cr & {\text{Relative maximum }}\left( { - 0.6812,0.4466} \right) \cr} $$

$$\eqalign{ & h\left( x \right) = \arcsin x - 2\arctan x \cr & {\text{Domain: }}\left[ { - 1,1} \right] \cr & {\text{Differentiate}} \cr & h'\left( x \right) = \frac{d}{{dx}}\left[ {\arcsin x - 2\arctan x} \right] \cr & h'\left( x \right) = \frac{1}{{\sqrt {1 - {x^2}} }} - \frac{2}{{1 + {x^2}}} \cr & {\text{Let }}y' = 0{\text{ to find critical points}} \cr & \frac{1}{{\sqrt {1 - {x^2}} }} - \frac{2}{{1 + {x^2}}} = 0 \cr & {\left( {\frac{1}{{\sqrt {1 - {x^2}} }}} \right)^2} = {\left( {\frac{2}{{1 + {x^2}}}} \right)^2} \cr & \frac{1}{{1 - {x^2}}} = \frac{4}{{{{\left( {1 + {x^2}} \right)}^2}}} \cr & {x^4} + 2{x^2} + 1 = 4 - 4{x^2} \cr & {x^4} + 6{x^2} - 3 = 0 \cr & {\text{By the quadratic formula}} \cr & {x^2} = \frac{{ - 6 \pm \sqrt {36 + 12} }}{2} = \frac{{ - 6 \pm \sqrt {48} }}{2} \cr & {x^2} = - 3 + 2\sqrt 3 \cr & x = \pm \sqrt {2\sqrt 3 - 3} \cr & x = - \sqrt {2\sqrt 3 - 3} ,{\text{ }}x = \sqrt {2\sqrt 3 - 3} \cr & \cr & *{\text{Find the second derivative}} \cr & y'' = \frac{d}{{dx}}\left[ {\frac{1}{{\sqrt {1 - {x^2}} }} - \frac{2}{{1 + {x^2}}}} \right] \cr & y'' = \frac{x}{{{{\left( {1 - {x^2}} \right)}^{3/2}}}} + \frac{{4x}}{{{{\left( {1 + {x^2}} \right)}^2}}} \cr & \cr & {\text{*Evaluate }}y''\left( x \right){\text{ at }}x = - \sqrt {2\sqrt 3 - 3} \cr & y''\left( { - \sqrt {2\sqrt 3 - 3} } \right) \approx - 3 < 0,{\text{Then there is a relative maximum}} \cr & {\text{at }}x = - \sqrt {2\sqrt 3 - 3} \approx - 0.6812 \cr & h\left( { - \sqrt {2\sqrt 3 - 3} } \right) \approx 0.4466 \cr & \cr & {\text{*Evaluate }}y''\left( x \right){\text{ at }}x = - \sqrt {2\sqrt 3 - 3} \cr & y''\left( {\sqrt {2\sqrt 3 - 3} } \right) \approx 3 > 0,{\text{Then there is a relative minimum}} \cr & {\text{at }}x = \sqrt {2\sqrt 3 - 3} \approx 0.6812 \cr & h\left( {\sqrt {2\sqrt 3 - 3} } \right) \approx - 0.4466 \cr & \cr & {\text{Relative minimum }}\left( {0.6812, - 0.4466} \right) \cr & {\text{Relative maximum }}\left( { - 0.6812,0.4466} \right) \cr & \cr & {\text{Graph}} \cr} $$