find extrema

Question

ive values and locations of all extrema of y=2−x

 where −8x2 (Note: Enter your answers as unordered sets, e.g.  x        

 {1,2,3})

x{}
y{}

Answer 1

The given function is  f(x) = y = 2^-|x| , In interval -8 < x ≤  2.

Apply first derivative with respect to x.

f '(x) = (2^-|x|)(log2)d/dx(|x|) = (2^-|x|)(log2)( x /|x| )

 

Find the critical points where f(x) does not exist or Set f '( x ) = 0

f '( x ) = 0  ⇒ (2^-|x|)(log2)( x /|x| )= 0  ⇒  x = 0.

Substitute critical point and end point of given interval (-8 < x ≤  2) to determine maxima or minima.

 

At x = 0 ⇒ f(0) = 2^-|0| = 2⁰ = 1

f(x) values at end points at given interval -8 < x ≤  2

At x = -8 ⇒ f(-8) = 2^-|-8| = 2^-8 = 0.00390625

At x = 2 ⇒ f(2) = 2^-|2| = 2^-2 = 0.25

From above three points

Maximum point occurs at x = 0  is 1.

Minimum point occurs at x = -8  is 0.00390625.

Solution :

Maximum point occurs at x = 0  is 1.

Minimum point occurs at x = -8  is 0.00390625.