Maharashtra Board 12th Commerce Maths Solutions Chapter 4 Applications of Derivatives Ex 4.3

Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf Chapter 4 Applications of Derivatives Ex 4.3 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 4 Applications of Derivatives Ex 4.3

Question 1.
Determine the maximum and minimum values of the following functions:
(i) f(x) = 2x3 – 21x2 + 36x – 20
Solution:
f(x) = 2x3 – 21x2 + 36x – 20
∴ f'(x) = \(\frac{d}{d x}\)(2x3 – 21x2 + 36x – 20)
= 2 × 3x2 – 21 × 2x + 36 × 1 – 0
= 6x2 – 42x + 36
and f”(x) = \(\frac{d}{d x}\)(6x2 – 42x + 36)
= 6 × 2x – 42 × 1 + 0
= 12x – 42
f'(x) = 0 gives 6x2 – 42x + 36 = 0.
∴ x2 – 7x + 6 = 0
∴ (x – 1)(x – 6) = 0
∴ the roots of f'(x) = 0 are x1 = 1 and x2 = 6.
For x = 1, f”(1) = 12(1) – 42 = -30 < 0
∴ by the second derivative test,
f has maximum at x = 1 and maximum value of f at x = 1
f(1) = 2(1)3 – 21(1)2 + 36(1) – 20
= 2 – 21 + 36 – 20
= -3
For x = 6, f”(6) = 12(6) – 42 = 30 > 0
∴ by the second derivative test,
f has minimum at x = 6 and minimum value of f at x = 6
f(6) = 2(6)3 – 21(6)2 + 36(6) – 20
= 432 – 756 + 216 – 20
= -128
Hence, the function f has maximum value -3 at x = 1 and minimum value -128 at x = 6.

Maharashtra Board 12th Commerce Maths Solutions Chapter 4 Applications of Derivatives Ex 4.3

(ii) f(x) = x . log x
Solution:
f(x) = x . log x
f'(x) = \(\frac{d}{d x}\)(x.log x)
= x.\(\frac{d}{d x}\)(log x) + log x.\(\frac{d}{d x}\)(x)
= x × \(\frac{1}{x}\) + (logx) × 1
= 1 + log x
and f”(x) = \(\frac{d}{d x}\)(1 + logx)
= 0 + \(\frac{1}{x}\)
= \(\frac{1}{x}\)
Now, f'(x) = 0, if 1 + log x = 0
i.e. if log x = -1 = -log e
i.e. if log x = log(e-1) = log \(\frac{1}{e}\)
i.e. if x = \(\frac{1}{e}\)
When x = \(\frac{1}{e}\), f”(x) = \(\frac{1}{(1 / e)}\) = e > 0
∴ by the second derivative test,
f is minimum at x = \(\frac{1}{e}\)
Minimum value of f at x = \(\frac{1}{e}\)
= \(\frac{1}{e}\) log(\(\frac{1}{e}\))
= \(\frac{1}{e}\) log(e-1)
= \(\frac{1}{e}\) (-1) log e
= \(\frac{-1}{e}\) ……..[∵ log e = 1]
Hence, the function f has minimum at x = \(\frac{1}{e}\) and minimum value is \(\frac{-1}{e}\).

(iii) f(x) = x2 + \(\frac{16}{x}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 4 Applications of Derivatives Ex 4.3 Q1(iii)
f'(x) = 0 gives 2x – \(\frac{16}{x^{2}}\) = 0
∴ 2x3 – 16 = 0
∴ x3 = 8
∴ x = 2
For x = 2, f”(2) = 2 + \(\frac{32}{(2)^{3}}\) = 6 > 0
∴ by the second derivative test, f has minimum at x = 2 and minimum value of f at x = 2
f(2) = (2)2 + \(\frac{16}{2}\)
= 4 + 8
= 12
Hence, the function f has a minimum at x = 2 and a minimum value is 12.

Maharashtra Board 12th Commerce Maths Solutions Chapter 4 Applications of Derivatives Ex 4.3

Question 2.
Divide the number 20 into two parts such that their product is maximum.
Solution:
Let the first part of 20 be x.
Then the second part is 20 – x.
∴ their product = x(20 – x) = 20x – x2 = f(x) …..(Say)
∴ f'(x) = \(\frac{d}{d x}\)(20x – x2)
= 20 × 1 – 2x
= 20 – 2x
and f”(x) = \(\frac{d}{d x}\)(20 – 2x)
= 0 – 2 × 1
= -2
The root of the equation f'(x) = 0
i.e. 20 – 2x = 0 is x = 10
and f”(10) = -2 < 0
∴ by the second derivative test, f is maximum at x = 10.
Hence, the required parts of 20 are 10 and 10.

Question 3.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions where its area is maximum.
Solution:
Let x cm and y cm be the length and breadth of the rectangle.
Then its perimeter is 2(x + y) = 36
∴ x + y = 18
∴ y = 18 – x
Area of the rectangle = xy = x(18 – x)
Let f(x) = x(18 – x) = 18x – x2
Then f'(x) = \(\frac{d}{d x}\)(18x – x2)
= 18 × 1 – 2x
= 18 – 2x
and f”(x) = \(\frac{d}{d x}\)(18 – 2x)
= 0 – 2 × 1
= -2
Now, f(x) = 0, if 18 – 2x = 0
i.e. if x = 9
and f”(9) = -2 < 0
∴ by the second derivative test, f has maximum value at x = 9
When x = 9, y = 18 – 9 = 9
Hence, the rectangle is a square of side 9 cm.

Maharashtra Board 12th Commerce Maths Solutions Chapter 4 Applications of Derivatives Ex 4.3

Question 4.
The total cost of producing x units is ₹(x2 + 60x + 50) and the price is ₹(180 – x) per unit. For what units is the profit maximum?
Solution:
Let the number of units sold be x.
Then profit = S.P. – C.P.
∴ P(x) = (180 – x)x – (x2 + 60x + 50)
∴ P(x) = 180x – x2 – x2 – 60x – 50
∴ P(x) = 120x – 2x2 – 50
P'(x) = \(\frac{d}{d x}\)(120x – 2x2 – 50)
= 120 × 1 – 2 × 2x – 0
= 120 – 4x
and P”(x) = \(\frac{d}{d x}\)(120 – 4x)
= 0 – 4 × 1
= -4
P'(x) = 0 if 120 – 4x = 0
i.e. if x = 30 and P”(30) = -4 < 0
∴ by the second derivative test, P(x) is maximum when x = 30.
Hence, the number of units sold for maximum profit is 30.

Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

Balbharti Maharashtra State Board 11th Maths Book Solutions Pdf Chapter 9 Differentiation Miscellaneous Exercise 9 Questions and Answers.

Maharashtra State Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

(I) Select the appropriate option from the given alternatives.

Question 1.
If y = \(\frac{x-4}{\sqrt{x+2}}\), then \(\frac{d y}{d x}\) is
(A) \(\frac{1}{x+4}\)
(B) \(\frac{\sqrt{x}}{\left(\sqrt{x+2)^{2}}\right.}\)
(C) \(\frac{1}{2 \sqrt{x}}\)
(D) \(\frac{x}{(\sqrt{x}+2)^{2}}\)
Answer:
(C) \(\frac{1}{2 \sqrt{x}}\)
Hint:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 I Q1

Question 2.
If y = \(\frac{a x+b}{c x+d}\),then \(\frac{d y}{d x}\) =
(A) \(\frac{a b-c d}{(c x+d)^{2}}\)
(B) \(\frac{a x-c}{(c x+d)^{2}}\)
(C) \(\frac{a c-b d}{(c x+d)^{2}}\)
(D) \(\frac{a d-b c}{(c x+d)^{2}}\)
Answer:
(D) \(\frac{a d-b c}{(c x+d)^{2}}\)
Hint:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 I Q2

Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

Question 3.
If y = \(\frac{3 x+5}{4 x+5}\), then \(\frac{d y}{d x}\) =
(A) \(-\frac{15}{(3 x+5)^{2}}\)
(B) \(-\frac{15}{(4 x+5)^{2}}\)
(C) \(-\frac{5}{(4 x+5)^{2}}\)
(D) \(-\frac{13}{(4 x+5)^{2}}\)
Answer:
(C) \(-\frac{5}{(4 x+5)^{2}}\)
Hint:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 I Q3

Question 4.
If y = \(\frac{5 \sin x-2}{4 \sin x+3}\), then \(\frac{d y}{d x}\) =
(A) \(\frac{7 \cos x}{(4 \sin x+3)^{2}}\)
(B) \(\frac{23 \cos x}{(4 \sin x+3)^{2}}\)
(C) \(-\frac{7 \cos x}{(4 \sin x+3)^{2}}\)
(D) \(-\frac{15 \cos x}{(4 \sin x+3)^{2}}\)
Answer:
(B) \(\frac{23 \cos x}{(4 \sin x+3)^{2}}\)
Hint:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 I Q4

Question 5.
Suppose f(x) is the derivative of g(x) and g(x) is the derivative of h(x).
If h(x) = a sin x + b cos x + c, then f(x) + h(x) =
(A) 0
(B) c
(C) -c
(D) -2(a sin x + b cos x)
Answer:
(B) c
Hint:
h(x) = a sin x + b cos x + c
Differentiating w.r.t. x, we get
h'(x) = a cos x – b sin x = g(x) …..[given]
Differentiating w.r.t. x, we get
g'(x) = -a sin x – b cos x = f(x) …..[given]
∴ f(x) + h(x) = -a sin x – b cos x + a sin x + b cos x + c
∴ f(x) + h(x) = c

Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

Question 6.
If f(x) = 2x + 6, for 0 ≤ x ≤ 2
= ax2 + bx, for 2 < x ≤ 4
is differentiable at x = 2, then the values of a and b are
(A) a = \(-\frac{3}{2}\), b = 3
(B) a = \(\frac{3}{2}\), b = 8
(C) a = \(\frac{1}{2}\), b = 8
(D) a = \(-\frac{3}{2}\), b = 8
Answer:
(D) a = \(-\frac{3}{2}\), b = 8
Hint:
f(x) = 2x + 6, 0 ≤ x ≤ 2
= ax2 + bx, 2 < x ≤ 4
Lf'(2) = 2, Rf'(2) = 4a + b
Since f is differentiable at x = 2,
Lf'(2) = Rf'(2)
∴ 2 = 4a + b …..(i)
f is continuous at x = 2.
∴ \(\lim _{x \rightarrow 2^{+}} f(x)=f(2)=\lim _{x \rightarrow 2^{-}} f(x)\)
∴ 4a + 2b = 2(2) + 6
∴ 4a + 2b = 10
∴ 2a + b = 5 …..(ii)
Solving (i) and (ii), we get
a = \(-\frac{3}{2}\), b = 8

Question 7.
If f(x) = x2 + sin x + 1, for x ≤ 0
= x2 – 2x + 1, for x ≤ 0, then
(A) f is continuous at x = 0, but not differentiable at x = 0
(B) f is neither continuous nor differentiable at x = 0
(C) f is not continuous at x = 0, but differentiable at x = 0
(D) f is both continuous and differentiable at x = 0
Answer:
(A) f is continuous at x = 0, but not differentiable at x = 0
Hint:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 I Q7

Question 8.
If f(x) = \(\frac{x^{50}}{50}+\frac{x^{49}}{49}+\frac{x^{48}}{48}+\ldots .+\frac{x^{2}}{2}+x+1\), then f'(1) =
(A) 48
(B) 49
(C) 50
(D) 51
Answer:
(C) 50
Hint:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 I Q8

Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

(II).

Question 1.
Determine whether the following function is differentiable at x = 3 where,
f(x) = x2 + 2, for x ≥ 3
= 6x – 7, for x < 3.
Solution:
f(x) = x2 + 2, x ≥ 3
= 6x – 7, x < 3
Differentiability at x = 3
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q1
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q1.1
Here, Lf'(3) = Rf'(3)
∴ f is differentiable at x = 3.

Question 2.
Find the values of p and q that make function f(x) differentiable everywhere on R.
f(x) = 3 – x, for x < 1
= px2 + qx, for x ≥ 1.
Solution:
f(x) is differentiable everywhere on R.
∴ f(x) is differentiable at x = 1.
∴ f(x) is continuous at x = 1.
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q2
f(x) is differentiable at x = 1.
∴ Lf'(1) = Rf'(1)
∴ -1 = 2p + q …..(ii)
Subtracting (i) from (ii), we get
p = -3
Substituting p = -3 in (i), we get
p + q = 2
∴ -3 + q = 2
∴ q = 5

Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

Question 3.
Determine the values of p and q that make the function f(x) differentiable on R where
f(x) = px3, for x < 2
= x2 + q, for x ≥ 2
Solution:
f(x) is differentiable on R.
∴ f(x) is differentiable at x = 2.
∴ f(x) is continuous at x = 2.
Continuity at x = 2:
f(x) is continuous at x = 2.
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q3
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q3.1
f(x) is differentiable at x = 2.
∴ Lf'(2) = Rf'(2)
∴ 12p = 4
∴ p = \(\frac{1}{3}\)
Substituting p = \(\frac{1}{3}\) in (i), we get
8(\(\frac{1}{3}\) – q = 4
∴ q = \(\frac{8}{3}\) – 4 = \(\frac{-4}{3}\)

Question 4.
Determine all real values of p and q that ensure the function
f(x) = px + q, for x ≤ 1
= tan(\(\frac{\pi x}{4}\)), for 1 < x < 2
is differentiable at x = 1.
Solution:
f(x) is differentiable at x = 1.
∴ f(x) is continuous at x = 1.
Continuity at x= 1:
f(x) is continuous at x = 1.
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q4
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q4.1

Question 5.
Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R.
Solution:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q5
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q5.1
Here, Lf'(1) ≠ Rf'(1)
∴ f is not differentiable at x = 1.
∴ f is not differentiable at x = -1 and x = 1.
∴ f is not differentiable ∀ x ∈ R.

Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

Question 6.
Test whether the function
f(x) = 2x – 3, for x ≥ 2
= x – 1, for x < 2
is differentiable at x = 2.
Solution:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q6

Question 7.
Test whether the function
f(x) = x2 + 1, for x ≥ 2
= 2x + 1, for x < 2
is differentiable at x = 2.
Solution:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q7
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q7.1

Question 8.
Test whether the function
f(x) = 5x – 3x2, for x ≥ 1
= 3 – x, for x < 1
is differentiable at x = 1.
Solution:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q8
Here, Lf'(1) = Rf'(1)
∴ f(x) is differentiable at x = 1.

Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9

Question 9.
If f(2) = 4, f'(2) = 1, then find \(\lim _{x \rightarrow 2}\left[\frac{x f(2)-2 f(x)}{x-2}\right]\)
Solution:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q9

Question 10.
If y = \(\frac{\mathbf{e}^{x}}{\sqrt{x}}\), find \(\frac{d y}{d x}\) when x = 1.
Solution:
Maharashtra Board 11th Maths Solutions Chapter 9 Differentiation Miscellaneous Exercise 9 II Q10