Calculus
Find the values for which f(x) = x^1010x 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.

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
posted by Damon

thank you so much, that makes so much more sense now!!!
posted by joe
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