Question

Find the values for which f(x) = x^10-10x and if 0 (less than or equal) to x (which is less than or equal to) 2.

a) f(x)has a local max or local min. Indicate which ones are max and which are mins.

b) f(x) has a global max or global min

could you kind of give me a hint on how to start and solve this question? i'm not really sure how to start working on it.

Answer 1

0 </= x </= 2 is the domain?
y = x^10 - 10 x
if you graph this it will start at 0, drop below axis, come back up to 0 when x^9 = 10 (about x = 1.3) and take off like mad up to the right
so part a
dy/dx = 10 x^9 - 10
zero when x^9 = 1
or when x = 1
that is in our domain and we already know it is a min but to prove it find y"
y" = 90 x^8 where x = 1
so y" is positive so this is a minimum at x = 1
there
y = 1^10 - 10 = -9
so
(1, -9) is local min
local max will be where x = 2 if y>0 there
y = 2^10 - 20
y = 1024 - 20 = 1004
so
(2, 1004) is the maximum in the domain

outside our domain x^10 is even so goes to oo for negative x and the -10x term is positive for negative x so it never drops below zero to the left of the origin and has no minima before heading off to +infinity for large -x

Answer 2

thank you so much, that makes so much more sense now!!!