76. f ( x ) = − x 3 + 12 x (a) Determine whether f is even, odd, or neither. (b) There is a local maximum value of 16 at 2. Determine the local minimum value.
76. f ( x ) = − x 3 + 12 x (a) Determine whether f is even, odd, or neither. (b) There is a local maximum value of 16 at 2. Determine the local minimum value.
Solution Summary: The author explains that the given function f is an odd function. The local minimum value is based on the local maximum value of 16 at 2.
(b) There is a local maximum value of 16 at 2. Determine the local minimum 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 72AYU
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 minimum value based on the local maximum value of 16 at 2.
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 72AYU
b. The local minimum value based on the local maximum value of 16 at 2 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 minimum value based on the local maximum value of 16 at 2.
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 maximum value of 16 at 2. Therefore, the local maximum point is .
The definition of odd function says that whenever the point is on the graph of , the point also on the graph.
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