Question

Find the absolute extrema if they​ exist, as well as all values of x where they​ occur, for the function f (x )equals2 x Superscript 4 Baseline minus 36 x squared plus 2 on the domain [negative 4 comma 4 ].
Find the derivative of f left parenthesis x right parenthesisequals2 x cubed plus 20 x squared plus 50 x plus 1. f prime left parenthesis x right parenthesisequals nothing Identify the absolute maximum if it​ exists, as well as all values of x where it occurs.

Answer 1

Step-by-step explanation:

Derivative of the first function:

$f(x)=2x^{4} -36x^{2} +2 on D:[-4,4]$

$f'(x)=8x^{3} -72x$

set that equal to 0 and solve for possible x values.

$8x(x^{2} -9)=0\\x=0, -3, 3$

put x back in to original equation to get a y value

y = 2, -160, -160,

Absolute min at (-3, -160) and (3, 160) and an absolute max at (0,2)

Second Part

Find derivative of $f(x)=2x^{3} +20x^{2} +50x +1$

$f'(x)=6x^{2} +40x+50$

set equal to zero, it is a quadratic so you can use the quadratic formula to solve for x

x= -5 or -5/3

put x back into the original function to get a y value

y = 1 or -973/27

so absolute max at (-5, 1) and absolute min at (-5/3, -973/27)