For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
Answer
The function has a minimum of y = 0. The minimum occurs at x = 1:
Example 2
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
Answer
The function has no minima. For any value of the function, there is a smaller value right next to it:
Example 3
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
Answer
The function has three minima: y = 3 at x = -1, y = 2 at x = 1, and y = 1 at x = 3:
Example 4
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
Answer
This is a sneaky one. Since the function always has a value of y = -2, and never takes on any smaller values, y = -2 is a minimum. The function takes on its minimum value everywhere!
Example 5
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
Answer
This function has no minima, since for any value of the function there is a smaller value nearby.
Example 6
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
f (x) = 2 – x2
Answer
This function has no minimum:
Example 7
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
f (x) = (x – 4)2 + 3
Answer
This function has a minimum of y = 3, since (x - 4)2 is always at least 0. The minimum occurs at x = 4.
Example 8
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
f (x) = 5
Answer
This function always takes the same value, y = 5. Since the function never becomes smaller than 5, 5 is a minimum. This minimum occurs everywhere.
Example 9
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
f (x) = 3x + 7
Answer
This is a line with positive slope. It has no minima, since as we move left the line is always decreasing.
Example 10
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
f (x) = sin x
Answer
The function f (x) = sin x has a minimum of -1. This minimum occurs at for all integers n.
Example 11
These questions deal with characteristics of the first and second derivatives of a function at and near a minimum value.
a. Below is a graph of a function f with a minimum at x = x0. Determine the sign of the derivative f ' at each labeled x-value.
b. Assume f is defined and twice differentiable on the whole real line. Around a minimum of the function f, is f concave up or concave down?
Answer
a. f ' is negative at x1 and x2, zero at x0, and positive at x3 and x4:
b. f is concave up, because a minimum occurs at the bottom of a right-side up bowl:
A maximum value of a function is a value of the function that is as high, or higher, than other nearby values of the function. A maximum looks like the top of a hill:
Example 12
These questions deal with characteristics of the first and second derivatives of a function at and near a maximum value.
a. Below is a graph of a function f with a maximum at x = x0. Determine the sign of the derivative f ' at each labeled x-value.
b. Assume f is defined and twice differentiable on the whole real line. Around a maximum of the function f, is f concave up or concave down?
Answer
a. f is positive at x1 and x2, zero at x0, and negative at x3 and x4:
b. f is concave down, because a maximum occurs at the top of an upside-down bowl:
for all integers n.
