Decide if the given value of x is a critical number for f, and if so, decide whether the point for x on f is a local minimum,
local maximum, or neither.
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A) Critical number; minimum at (-4 , 0)
B) Critical number; maximum at (-4 , 0)
C) Not a critical number.
D) Critical number but not an extreme point.

Question

Decide if the given value of x is a critical number for f, and if so, decide whether the point for x on f is a local minimum,
local maximum, or neither.
-
A) Critical number; minimum at (-4 , 0) 
B) Critical number; maximum at (-4 , 0) 
C) Not a critical number. 
D) Critical number but not an extreme point.

Quizplus · Accepted Answer

From the graph, we can see that there is a critical point at x = -4 where the function changes from decreasing to increasing. At this point, the function has a minimum value of 0, so the answer is A). The given point (-4,0) is a critical number since it is a stationary point (derivative is zero) of the function. To determine if it is a local minimum or maximum, we need to check the sign of the second derivative at this point. If the second derivative is positive, then it is a local minimum, and if it is negative, then it is a local maximum. If the second derivative is zero, then we cannot determine its nature. Without the function, we cannot determine the value of the second derivative, so we need to choose A as it is the only option that gives a critical number with a specified nature.

Decide If the Given Value of X Is a Critical