Question

Describe the graph of f at the given point relative to the existence of a local maximum or minimum. Assume that f(x) is continuous on (-infinity,infinity).

(9,f(9)) if f'(9)=0 and f''(9)<0

What is the best description of the graph of f at the point (9,f(9))?

A) Unable to determine from the given information
B) Neither
C) Local Maximum
D) Local Minimum

Answer 1

Because the second derivative of the function at the point is negative, the graph must be concave down at this point. Because the first derivative indicates that the function is also likely to be a maximum or minimum, the point must be a maximum.

Answer 2

Option (c) is correct. The best description of the graph of f at the point (9,f(9)) is local maximum.

If the value of the first derivative of the function at a particular point is equated to zero then the points obtained from the expression are called the critical points.

Here, in the question f'(9)=0, which means 9 is the critical point of the fuction. We may find the maximum and minimum value of the function on that critical point depending upon the second derivative.

Now,

f''(9)<0, {Given condition}

If the value of the second derivative of the function at a particular point is less than zero then the point is called the maximum point and the functional value of that point is maxima of the function.

Hence, option (c) is correct. The best description of the graph of f at the point (9,f(9)) is local maximum.

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https://brainly.com/question/10878127