Question

PRECALCULUS : Extreme Value Theorem

im confuse could someone help me understand this

Answer 1

@mukushla @hartnn

Answer 2

Hey, that's Calculus, not precalculus.

Answer 3

my notes has :

The Extreme Value Theorem

Theorem: If a function, f(x), is continuous on a closed interval [a, b], then f(x) has both a maximum and a minimum on [a, b].

The theorem is stating that if you graph a curve from x = a and never pick your pencil up until x = b, then that curve is guaranteed to have a maximum (highest y value on graph) and a minimum value (lowest y value on graph). To identify where a maximum or minimum might occur, we have to consider two types of test values:

x values at the endpoints of the curve, x = a and x = b

any x value where the curve hits a peak (slope changes from positive to negative) or a valley (slope changes from negative to positive)
Once you identify the test points, you must substitute each x value into the function to determine its y value. Knowing the function’s y values at these test values will allow you to identify the maximum and minimum on the curve.

Let’s look at an example together. Graph f(x) = 4x – x2 on the interval [–1, 4]. Make sure you change your window settings on your calculator so that your x min is at –1 and x max is at 4. Your picture should look like this:

Answer 4

lol i guess it is, i have it in polynomials module :(

Answer 5

so whats the doubt ?

Answer 6

like ive understood intermediate value theorem and its use, i just dont get this at all it looks confusing i dont understand what its saying

Answer 7

i can test the function at endpoints. how should i know the valley and peaks ?

Answer 8

can u differentiate ?

Answer 9

ok thats calculus i believe i may have to just graph and see for now in precal ?

Answer 10

yes, clope changes from positive to negative...does this make sense ?

Answer 11

*slope

Answer 12

yes it does only thing is graph doesnt give me exact x coordinate where the valley or peak occurs... i have to zoom in and approximate always :(

Answer 13

yes, at that time slope =0<---horizontal line

Answer 14

i have another question on extreme value theorem

Answer 15

like if my graph is just a horizontal straight line, clearly it doesnt have

Answer 16

highest or lowest points... does extreme value theorem fails here ?

Answer 17

its value is fixed or constant in the given range [A, B] -> n

just a spur of the moment question sorry

Answer 18

highest point = lowest point = all the points = infinite number of points.

Answer 19

lol ok thnks :P

Answer 20

any more doubts... i think i have not explained well.....

Answer 21

no it makes very good sense thank you i get it :)

Answer 22

ok :)