Function similar to f x = 1 2 π e − x 2 / 2 arise in a wide variety of statistical problems. (a) Use the first derivate test to show that f has a relative maximum at x = 0 , and confirm this by using a graphing utility of graph f . (b) Sketch the graph of f x = 1 2 π e − x − μ 2 / 2 where μ is a constant, and label the coordinates of the relative extrema.
Function similar to f x = 1 2 π e − x 2 / 2 arise in a wide variety of statistical problems. (a) Use the first derivate test to show that f has a relative maximum at x = 0 , and confirm this by using a graphing utility of graph f . (b) Sketch the graph of f x = 1 2 π e − x − μ 2 / 2 where μ is a constant, and label the coordinates of the relative extrema.
Function similar to
f
x
=
1
2
π
e
−
x
2
/
2
arise in a wide variety of statistical problems.
(a) Use the first derivate test to show that
f
has a relative maximum at
x
=
0
,
and confirm this by using a graphing utility of graph
f
.
(b) Sketch the graph of
f
x
=
1
2
π
e
−
x
−
μ
2
/
2
where
μ
is a constant, and label the coordinates of the relative extrema.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
Example 1.5
Calculate the maximum absolute and relative error for function
xy?
f(x,y,z) = :
where
x = 2+ 0.003,
y = 3+0.0005,
z = 1+ 0.004
Solution
Ax = 0.003, Ay = 0.0005 ,
Az = 0.004
Review
Question 22
Consider the following function:
where
f(x) = 2x² — x³ + 1
€ [-1, 3]
x
(a) Find the x that are critical value points of f(x) (values of x that are a potential mini-
mum or maximum).
(b) Are those critical values maximum or minimum? (Hint: Check the second order
derivative. Check also the value of the function at the endpoints -1 and ½ to make
sure you have the correct answer.)
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