75. g ( x ) = x 3 − 27 x (a) Determine whether g is even, odd, or neither. (b) There is a local minimum value of − 54 at 3. Determine the local maximum value.
75. g ( x ) = x 3 − 27 x (a) Determine whether g is even, odd, or neither. (b) There is a local minimum value of − 54 at 3. Determine the local maximum value.
(b) There is a local minimum value of
at 3. Determine the local maximum value.
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 .
Expert Solution
To determine
Check whether the function is even, odd or neither.
Answer to Problem 71AYU
a. The given function is an odd function.
Explanation of Solution
Given:
The function .
Calculation:
It is asked to check the whether the function is even, odd or neither and find the local maximum value based on the local minimum value of at 3.
By the definition of odd and even function,
“A function is even if, for every number in its domain, the number is also in the domain and ” and
“A function is odd if, for every number in its domain, the number is also in the domain and ”.
“A function is odd if and only if, whenever the point is on the graph of , the point is also on the graph.
a. Consider the function,
Replace by ,
.
From the statement, it can be concluded that the given function is an odd function.
Expert Solution
To determine
Check whether the function is even, odd or neither.
Answer to Problem 71AYU
b. The local maximum value based on the local minimum value of at 3 is .
Explanation of Solution
Given:
The function .
Calculation:
It is asked to check the whether the function is even, odd or neither and find the local maximum value based on the local minimum value of at 3.
By the definition of odd and even function,
“A function is even if, for every number in its domain, the number is also in the domain and ” and
“A function is odd if, for every number in its domain, the number is also in the domain and ”.
“A function is odd if and only if, whenever the point is on the graph of , the point is also on the graph.
b. There is a local minimum value of at 3. Therefore, the local minimum point is .
The definition of odd function says that whenever the point is on the graph of , the point also on the graph.
Thomas' Calculus: Early Transcendentals (14th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.