Mathematics Lab Activity-07 Class XII

Mathematics Laboratory Activities on Applications of Derivatives for class XI Non-Medical students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 06 

applications of derivatives

Activity - 07

Objective

To understand the concept of local maxima, local minima and point of inflection

Material Required

A piece of plywood, wires, adhesive, white paper etc.

Procedure

1. Take a plywood of convenient size and paste a white paper on it.

2. Take two pieces of wires each of length 40 cm and fix them on the paper on plywood in the form of x-axis and y-axis.

3. Take another wire of suitable length and bend it in the shape of curve. Fix this curved wire on the white paper pasted on plywood, as shown in figure 7.1

Figure 7.1

4. Take five more wires each of length 2 cm and fix them at the points A, C, B, P and D as shown in the figure 7.1

5. In the fig. 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. The tangent at P intersects the curve.

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

7. At the points C and D, sign of first derivative changes from positive to negative. So, they are the points of local maxima.

8. At the point P, sign of first derivative does not change. So it is the point of inflection.

Observations

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

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

3. Sign of the slope of the tangent (first derivative) at a point on the curve to the immediate left of B is - ve.

4. Sign of the slope of the tangent (first derivative) at a point on the curve to the immediate right of B is + ve.

5. Sign of first derivative at a point on the curve to the immediate left of C is + ve.

6. Sign of first derivative at a point on the curve to the immediate right of C is - ve.

7. Sign of first derivative at a point on the curve to the immediate left of D is + ve.

8. Sign of first derivative at a point on the curve to the immediate right of D is - ve.

9. Sign of first derivative at a point on the curve to the immediate left of P is + ve and immediate right of P is – ve.

10. A and B are the point of local minima.

11. C and D are the point of Local Maxima.

12. Point P is called the point of inflection.

Result

If first derivative changes its sign from –ve to +ve then the function have local minimum value.

If first derivative changes its sign from +ve to -ve then the function have local maximum value.

If at any point first derivative does not change its sign then  that point is called the point of inflection.

Applications

This activity is useful in explaining the concept of local maxima, local minima and point of inflection.

VIVA – VOICE

Q1. Define local maxima?

Ans. A function 'f' is said to have local maxima at point x = c, if there exist h > 0, such that f(c) ≥ f(x) ∀ x ∈ (c-h, c+h).

Q2. Define local minima ?

Ans. A function 'f' is said to have local minima at point x = c, if there exist h > 0, such that f(c) ≤ f(x) ∀ x ∈ (c-h, c+h).

Q3. Is local maxima is always absolute maxima ?

Ans. No.

Q4. Is local minima is always absolute minima ?

Ans. No.

Q5. For local maxima or minima, the derivative of the function is not necessarily zero. Is this true ?

Ans. No.

Q6. How can be get critical points ?

Ans. By putting f'(x) = 0.

Q7. What is another name of the critical point ?

Ans. Stationary point.

Q8. What is the sign of f''(x) for local maxima ?

Ans. f''(x) is negative.

Q9. What is the sign of f''(x) for local minima ?

Ans. f''(x) is positive.

Q10. If f'(x) changes its sign from -ve to +ve at particular, then the function have point of local maxima. Is it true ?

Ans. No.

THANKS FOR YOUR VISIT

PLEASE COMMENT BELOW