Question

The derivative of the curve

• A. Will be same at all the points
• B. Will be positive at all points
• C. Will be negative at all points
• D. Will be positive at some points and negative at some points and zero at some

Answer 1

The correct option is D Will be positive at some points and negative at some points and zero at some

We know that for a straight line the increase in y for a particular increase in x is same throughout. That’s why the curve is a straight line. This curve, not being a straight line, will have slopes of tangents which will be different at different points, let’s say at A, B, C and D.

So slope of tangent will be different at different points which is nothing but the derivative of the function at that point.
Also we already discussed that when value of y increases with increase in value of x then the slope of the tangent ( derivative ) is positive and if it decreases with increase in value of x, then its negative. Here the curve increases and decreases for different portions with increase in value of x. It increases till A, then from A it decreases till C and then increases again. So slope or derivative will be negative at some points and positive at others and zero at some other points where the function does not change with x.
The slope is zero at the points where the function neither increases nor decreases i.e. remains constant.