SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.4 - Applications of Derivatives [Latest edition]

The slope of the tangent to the curve y = x³ – x² – 1 at the point whose abscissa is – 2, is ______.

Choose the correct alternative:

Slope of the normal to the curve 2x² + 3y² = 5 at the point (1, 1) on it is 

Choose the correct alternative:

The function f(x) = x³ – 3x² + 3x – 100, x ∈ R is

If the marginal revenue is 28 and elasticity of demand is 3, then the price is ______.

The price P for the demand D is given as P = 183 + 120D − 3D², then the value of D for which price is increasing, is ______.

If the elasticity of demand η = 1, then demand is ______.

If 0 < η < 1, then the demand is ______.

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

If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______

The slope of the tangent to the curve x = `1/"t"`, y = `"t" - 1/"t"`, at t = 2 is ______

If the average revenue is 45 and elasticity of demand is 5, then marginal revenue is ______.

The total cost function for production of articles is given as C = 100 + 600x – 3x², then the values of x for which the total cost is decreasing is  ______

State whether the following statement is True or False:

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

State whether the following statement is True or False: 

The function f(x) = `3/x` + 10, x ≠ 0 is decreasing

The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing

State whether the following statement is True or False:

The equation of tangent to the curve y = x² + 4x + 1 at (– 1, – 2) is 2x – y = 0 

State whether the following statement is True or False: 

If the function f(x) = x² + 2x – 5 is an increasing function, then x < – 1

State whether the following statement is True or False:  

If the marginal revenue is 50 and the price is ₹ 75, then elasticity of demand is 4

Find the equations of tangent and normal to the curve y = 3x² – x + 1 at the point (1, 3) on it

Find the values of x such that f(x) = 2x³ – 15x² + 36x + 1 is increasing function

Find the values of x such that f(x) = 2x³ – 15x² – 144x – 7 is decreasing function

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

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

If the demand function is D = 50 – 3p – p². Find the elasticity of demand at p = 5 comment on the result

If the demand function is D = 50 – 3p – p². Find the elasticity of demand at p = 2 comment on the result

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

The total cost of manufacturing x articles is C = 47x + 300x² − x⁴.  Find x, for which average cost is increasing.

The total cost of manufacturing x articles C = 47x + 300x² – x⁴ . Find x, for which average cost is decreasing

Determine the maximum and minimum value of the following function.

f(x) = 2x³ – 21x² + 36x – 20

A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.

Find MPC, MPS, APC and APS, if the expenditure E_c of a person with income I is given as E_c = (0.0003) I² + (0.075) I ; When I = 1000.

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

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 profit is increasing

If x + y = 3 show that the maximum value of x²y is 4.

Find the equation of tangent to the curve x² + y² = 5, where the tangent is parallel to the line 2x – y + 1 = 0

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

Find the equation of tangent to the curve y = x² + 4x at the point whose ordinate is – 3

A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.

Solution: Let the dimensions of the rectangle be x cm and y cm.

∴ 2x + 2y = 36

Let f(x) be the area of rectangle in terms of x, then

f(x) = `square`

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme value, f'(x) = 0, we get

x = `square`

∴ f''`(square)` = – 2 < 0

∴ Area is maximum when x = `square`, y = `square`

∴ Dimensions of rectangle are `square`

By completing the following activity, examine the function f(x) = x³ – 9x² + 24x for maxima and minima

Solution: f(x) = x³ – 9x² + 24x

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme values, f'(x) = 0, we get

x = `square` or `square`

∴ f''`(square)` = – 6 < 0

∴ f(x) is maximum at x = 2.

∴ Maximum value = `square`

∴ f''`(square)` = 6 > 0

∴ f(x) is maximum at x = 4.

∴ Minimum value = `square`

By completing the following activity, find the values of x such that f(x) = 2x³ – 15x² – 84x – 7 is decreasing function.

Solution: f(x) = 2x³ – 15x² – 84x – 7

∴ f'(x) = `square`

∴ f'(x) = 6`(square) (square)`

Since f(x) is decreasing function.

∴ f'(x) < 0

Case 1: `(square)` > 0 and (x + 2) < 0

∴ x ∈ `square`

Case 2: `(square)` < 0 and (x + 2) > 0

∴ x ∈ `square`

∴ f(x) is decreasing function if and only if x ∈ `square`

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

Solution: Total cost C = 40 + 2x and Price p = 120 – x

Revenue R = `square`

Differentiating w.r.t. x,

∴ `("dR")/("d"x) = square`

Since Revenue is increasing,

∴ `("dR")/("d"x)` > 0

∴ Revenue is increasing for `square`

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

Solution: Total cost C = 40 + 2x and Price p = 120 − x

Profit π = R – C

∴ π = `square`

Differentiating w.r.t. x,

`("d"pi)/("d"x)` = `square`

Since Profit is increasing,

`("d"pi)/("d"x)` > 0

∴ Profit is increasing for `square`

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

Solution: Total cost C = 40 + 2x and Price p = 120 – x

p = 120 – x

∴ x = 120 – p

Differentiating w.r.t. p,

`("d"x)/("dp")` = `square`

∴ Elasticity of demand is given by η = `- "P"/x*("d"x)/("dp")`

∴ η = `square`

When p = 80, then elasticity of demand η = `square`