Example 1
For the function, (a) find all minima of the function, if any, and (b) determine the x-value(s) at which each minima occurs.
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.
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.
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.
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.
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
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
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
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
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
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?
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?
