Question

If the derivative of a function f(x) is negative at a particular point, then about that point, the value of the function as the x-value increases.

• A. decreases
• B. increases
• C. remains the same

Answer 1

The correct option is A decreases

Look at the following curves carefully.

We know that the derivative at a point is nothing but the slope of tangent at that point which in itself is indicative of the steepness of the curve at that point. In curve (1) it is easy to observe that at every point (just like point A), the slope of the tangent (tan ɵ) should be positive and the curve , you can see is increasing as value of x-increases. But in curve (2)you can see that the tan ɵ (slope of tangent) is always negative (because ɵ here is always obtuse) and the curve is decreasing as the x-value increases.
Since $tan\theta$ = derivative of f(x). So if derivative is negative, then, y value decreases as x – value increases.