Question

Find any stationary points of the graph of
`y = 2x^2 + e^-x^4`

Answer 1

`y = 2x {}^(2) + e {}^{ - x {}^(4) }`

`(dy)/(dx) = 4x +( e {}^{ - x {}^(4) } )( - 4x {}^(3) )`

`(dy)/(dx) = 4x - 4x {}^(3) e { }^{ - x {}^(4) }`

`(dy)/(dx) = 4x(1 - x {}^(2) e {}^{ - x {}^(4) } )`

`(dy)/(dx) = 0 \\ 4x = 0 \: \: \: \: \: 1 - x {}^(2) e {}^{ - x {}^(4) } = 0`

`4x = 0 \\ x = 0 \: \: \: \: \: \: y(0) = 0 + e {}^(0) = 1`

`\frac{x {}^(2) }{e {}^{x {}^(4) } } = 1 \\ x {}^(2) = e {}^{x {}^(4) } \\ 2ln(x) = x {}^(4) \\ these \: 2 \: functions \: do \: not \: intersect`

Stationary point ( 0 , 1 )