Is the Extrema Global or Local?

Question

(Calculator allowed)

Is there a global minimum, a local minimum, both, or neither on the function f(x) = 4(x-1)⁴+2x²-3 at x = 0.5?

Answer 1

y= 4(x-1)^4 +2x^2-3

y(.5) = 4(.5-1)^4 +2(.5)^2 -3

= 4(-.5)^4 + 2(.25)-3

= 4(.0625) + .5-3

= .25 - 2.5

= - 2.25 = - 2 1/4

if you found the function's value when x is "huge" you'd get a large positive number greater than -2.25, so it's not a global maximum

question is can you get a smaller value for the function at different x value?

try x=1, the 1st term =0, and what remains is 2(1)-3 =-1 but that's greater than - 2.25

so good chance it's a global minimum

use a graphing calculator and see if the graph ever goes lower than -2.25

the graph looks a little bit like an upward opening parabola, with vertex = global minimum

from the graph, it is a global and local minimum at (.5, -2.25)