The graph below is the graph of a. Is the graph increasing or decreasing at x =

Question

The graph below is the graph of

a. Is the graph increasing or decreasing at x = 0? How can you tell from the sign of the derivative?

b. Is the graph increasing or decreasing at x = 7? How can you tell from the sign of the derivative?

c. Is the graph increasing or decreasing at x = 3? How can you tell from the value of the derivative?

The graph below is the graph of a. Is the graph increasing or decreasing at x = 0? How can you tell from the sign of the derivative? b. Is the graph increasing or decreasing at x = 7? How can you tell from the sign of the derivative? c. Is the graph increasing or decreasing at x = 3? How can you tell from the value of the derivative?

Explanation / Answer

Since the graph is not visible, i am gonna just explain it to you

I am assuming that the given graph is the graph of f'(x), the derivative

Wherever the value of f'(x) is negative, those are the regions where the f(x) will be DECREASING
And wherever the value of f'(x) is positive, those are the regions where the function, f(x) is INCREASING

Now, i assume that at x = 0 and x = 3, the f'(x) does not intersect the x-axis.

So, basically, just by telling from the sign of f'(x), we can tell.

If at x = 0, f'(x) is positive, then at x = 0, the function, f(x) is INCREASING
If at x = 0, f'(x) is a negative value, then at x = 0, the function f(x) will be DECREASING

Same for part B

Now, part C asks "how can we tell from the VALUE of the derivative"

From this, i assume that at x = 3, f'(x) = 0.
Now, since f'(x) is 0, the graph of f(x) is neither increasing nor decreasing as the point x = 3 is a relative extreme point, Now, this rel ext point can be a maxima or a minima

If the f'(x) went from a positive to negative value at x = 3, then the point x = 3 is a relative MAXIMUM
IF the f'(x) went from a negative to positive value at x = 3, then the point x = 3 is a relative MINIMUM

I hope this cleared it up for you

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