Page 57 - Mathematics Class - XII

DEMONSTRATION
1. In the figure, wires at the points A, B, C and D represent tangents to the curve and are parallel to the axis.
The slopes of tangents at these points are zero, i.e., the value of the first derivative at these points is zero.
The tangent at P intersects the curve.

2. At the points A and B, sign of the first derivative changes from negative to positive. So, they are the points
of local minima.

3. At the point C and D, sign of the first derivative changes from positive to negative. So, they are the points
of local maxima.
4. At the point P, sign of first derivative does not change. So, it is a point of inflection.

OBSERVATION
1. Sign of the slope of the tangent (first derivative) at a point on the curve to the immediate left of A is
negative.

2. Sign of the slope of the tangent (first derivative) at a point on the curve to the immediate right of A is
positive.

3. Sign of the first derivative at a point on the curve to immediate left of B is negative.
4. Sign of the first derivative at a point on the curve to immediate right of B is positive.

5. Sign of the first derivative at a point on the curve to immediate left of C is positive.
6. Sign of the first derivative at a point on the curve to immediate right of C is negative.

7. Sign of the first derivative at a point on the curve to immediate left of D is positive.
8. Sign of the first derivative at a point on the curve to immediate right of D is negative.

9. Sign of the first derivative at a point immediate left of P is positive and immediate right of P is positive.

10. A and B are points of local minima.
11. C and D are points of local maxima.

12. P is a point of inflection.

CONCLUSION
In the above activity, the concepts of local maxima, local minima and point of inflection have been
demonstrated.

APPLICATION
This activity may help in explaining the concepts of points of local maxima, local minima and inflection.

Knowledge Booster

1. If f ′(x) = 0 at a point, and if f ˝(x) > 0 there, then that point must be a minimum.
2. If f ′(x) = 0 at a point, and if f ˝(x) < 0 there, then that point must be a maximum.

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