#### Chapters

## Chapter 4: Applications of Derivatives

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.1 [Page 105]

**Find the equation of tangent and normal to the curve at the given points on it.**

y = 3x^{2} - x + 1 at (1, 3)

**Find the equation of tangent and normal to the curve at the given points on it.**

2x^{2} + 3y^{2} = 5 at (1, 1)

**Find the equation of tangent and normal to the curve at the given points on it.**

x^{2} + y^{2} + xy = 3 at (1, 1)

Find the equations of tangent and normal to the curve y = x^{2} + 5 where the tangent is parallel to the line 4x − y + 1 = 0.

Find the equations of tangent and normal to the curve y = 3x^{2} - 3x - 5 where the tangent is parallel to the line 3x − y + 1 = 0.

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.2 [Page 106]

Test whether the following functions are increasing or decreasing : f(x) = x^{3} – 6x^{2} + 12x – 16, x ∈ R.

**Test whether the function is increasing or decreasing.**

f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,

**Test whether the following function is increasing or decreasing.**

f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0

**Find the value of x, such that f(x) is increasing function.**

f(x) = 2x^{3} - 15x^{2} + 36x + 1

**Find the value of x, such that f(x) is increasing function.**

f(x) = x^{2} + 2x - 5

**Find the value of x, such that f(x) is increasing function.**

f(x) = 2x^{3} - 15x^{2} - 144x - 7

**Find the value of x, such that f(x) is decreasing function.**

f(x) = 2x^{3} - 15x^{2} - 144x - 7

**Find the value of x such that f(x) is decreasing function.**

f(x) = x^{4} − 2x^{3} + 1

**Find the value of x, such that f(x) is decreasing function.**

f(x) = 2x^{3} - 15x^{2} - 84x - 7

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.3 [Page 109]

**Determine the maximum and minimum value of the following function.**

f(x) = 2x^{3} - 21x^{2} + 36x - 20

**Determine the maximum and minimum value of the following function.**

f(x) = x log x

**Determine the maximum and minimum value of the following function.**

f(x) = `"x"^2 + 16/"x"`

Divide the number 20 into two parts such that their product is maximum.

A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.

The total cost of producing x units is ₹ (x^{2} + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Exercise 4.4 [Pages 112 - 113]

The demand function of a commodity at price P is given as, D = `40 - "5P"/8`. Check whether it is increasing or decreasing function.

Price P for demand D is given as P = 183 +120D - 3D^{2} Find D for which the price is increasing

The total cost function for production of x articles is given as C = 100 + 600x – 3x^{2} . Find the values of x for which total cost is decreasing.

The manufacturing company produces x items at the total cost of ₹ 180 + 4x. The demand function for this product is P = (240 – x). Find x for which revenue is increasing.

A manufacturing company produces x items at the total cost of Rs (180+4x). The demand function of this product is P=(240 - x). Find x for which profit is increasing.

For manufacturing x units, labour cost is 150 - 54x, processing cost is x^{2} and revenue R = 10800x - 4x^{3 }. Find the value of x for which Total cost is decreasing.

For manufacturing x units, labour cost is 150 – 54x and processing cost is x^{2}. Price of each unit is p = 10800 – 4x^{2}. Find the values of x for which Revenue is increasing.

The total cost of manufacturing x articles is C = (47x + 300x^{2} - x^{4}). Find x, for which average cost is increasing.

The total cost of manufacturing x articles C = 47x + 300x^{2} – x^{4} . Find x, for which average cost is decreasing.

Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.

Find the price, if the marginal revenue is 28 and elasticity of demand is 3.

Find the elasticity of demand, if the marginal revenue is 50 and price is Rs 75.

If the demand function is D = `(("p" + 6)/("p" - 3))`, find the elasticity of demand at p = 4.

Find the price for the demand function D = `((2"p" + 3)/(3"p" - 1))`, when elasticity of demand is `11/14`.

If the demand function is D = 50 – 3p – p^{2}. Find the elasticity of demand at p = 5 comment on the result.

If the demand function is D = 50 – 3p – p^{2}. Find the elasticity of demand at p = 2 comment on the result.

For the demand function D = 100 – `"p"^2/2`. Find the elasticity of demand at p = 10 and comment on the results.

For the demand function D = 100 – `"p"^2/2`. Find the elasticity of demand at p = 6 and comment on the results.

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price is given as p = 120 – x. Find the value of x for which revenue is increasing.

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price is given as p = 120 – x. Find the value of x for which profit is increasing.

A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price is given as p = 120 – x. Find the value of x for which also find an elasticity of demand for price 80.

Find MPC, MPS, APC and APS, if the expenditure E_{c} of a person with income I is given as E_{c} = (0.0003) I^{2} + (0.075) I When I = 1000.

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Applications of Derivatives Miscellaneous Exercise 4 [Pages 113 - 114]

**Choose the correct alternative.**

The equation of tangent to the curve y = x^{2} + 4x + 1 at (-1, -2) is

2x - y = 0

2x + y - 5 = 0

2x - y - 1 = 0

x + y - 1 = 0

**Choose the correct alternative.**

The equation of tangent to the curve x^{2} + y^{2} = 5 where the tangent is parallel to the line 2x – y + 1 = 0 are

2x – y + 5 = 0; 2x – y – 5 = 0

2x + y + 5 = 0; 2x + y – 5 = 0

x – 2y + 5 = 0; x – 2y – 5 = 0

x + 2y + 5 = 0; x + 2y – 5 = 0

**Choose the correct alternative.**

If elasticity of demand η = 1, then demand is

constant

inelastic

unitary elastic

elastic

**Choose the correct alternative.**

If 0 < η < 1, then demand is

constant

inelastic

unitary elastic

elastic

**Choose the correct alternative.**

The function f(x) = x^{3} - 3x^{2} + 3x - 100, x ∈ R is

increasing for all x ∈ R, x ≠ 1

decreasing

neither, increasing nor decreasing

decreasing for all x ∈ R, x ≠ 1

**Choose the correct alternative.**

If f(x) = 3x^{3} - 9x^{2} - 27x + 15 then

f has maximum value 66

f has minimum value 30

f has maxima at x = –1

f has minima at x = –1

**Fill in the blank:**

The slope of tangent at any point (a, b) is called as _______.

**Fill in the blank:**

If f(x) = x - 3x^{2} + 3x - 100, x ∈ R then f''(x) is ______

**Fill in the blank:**

If f(x) = `7/"x" - 3`, x ∈ R x ≠ 0 then f ''(x) is ______

**Fill in the blank:**

A road of 108 m length is bent to form a rectangle. If the area of the rectangle is maximum, then its dimensions are _______.

**Fill in the blank:**

If f(x) = x log x, then its minimum value is______

**State whether the following statement is True or False:**

The equation of tangent to the curve y = 4xe^{x} at `(-1, (- 4)/"e")` is ye + 4 = 0

True

False

**State whether the following statement is True or False:**

x + 10y + 21 = 0 is the equation of normal to the curve y = 3x^{2} + 4x - 5 at (1, 2).

True

False

**State whether the following statement is True or False:**

An absolute maximum must occur at a critical point or at an end point.

True

False

**State whether the following statement is True or False:**

The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.

True

False

**Find the equation of tangent and normal to the following curve.**

xy = c^{2} at `("ct", "c"/"t")` where t is parameter.

**Find the equation of tangent and normal to the following curve.**

y = x^{2} + 4x at the point whose ordinate is -3.

**Find the equation of tangent and normal to the following curve.**

x = `1/"t", "y" = "t" - 1/"t"`, at t = 2

**Find the equation of tangent and normal to the following curve.**

y = x^{3} - x^{2} - 1 at the point whose abscissa is -2.

Find the equation of tangent to the curve y = `sqrt("x - 3")` which is perpendicular to the line 6x + 3y – 4 = 0.

Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.

Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.

If x + y = 3 show that the maximum value of x^{2}y is 4.

6. Examine the function for maxima and minima f(x) = x^{3} - 9x^{2} + 24x

## Chapter 4: Applications of Derivatives

## Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 - Applications of Derivatives

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Concepts covered in Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 Applications of Derivatives are Introduction of Derivatives, Increasing and Decreasing Functions, Maxima and Minima, Application of Derivatives to Economics.

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