Step 1: Definition of a point of inflection.
A point of inflection is a point where the concavity of the function changes. This means \( f''(x) \) changes its sign at that point.
Step 2: Conditions for inflection points.
At a point \( x = c \), the function \( f(x) \) has a point of inflection if: - \( f''(x) \) changes its sign around \( x = c \). - \( f'(x) = 0 \) but does not change its sign.
Step 3: Analyzing the options.
Option (C) correctly states that \( f'(x) = 0 \) and \( f'(x) \) does not change its sign, which is consistent with the definition of a point of inflection.
Step 4: Conclusion.
The correct answer is (C).
List-I | List-II |
(A) Absolute maximum value | (I) 3 |
(B) Absolute minimum value | (II) 0 |
(C) Point of maxima | (III) -5 |
(D) Point of minima | (IV) 4 |
In number theory, it is often important to find factors of an integer \( N \). The number \( N \) has two trivial factors, namely 1 and \( N \). Any other factor, if it exists, is called a non-trivial factor of \( N \). Naresh has plotted a graph of some constraints (linear inequations) with points \( A(0, 50) \), \( B(20, 40) \), \( C(50, 100) \), \( D(0, 200) \), and \( E(100, 0) \). This graph is constructed using three non-trivial constraints and two trivial constraints. One of the non-trivial constraints is \( x + 2y \geq 100 \).
Based on the above information, answer the following questions:
On her birthday, Prema decides to donate some money to children of an orphanage home.
If there are 8 children less, everyone gets ₹ 10 more. However, if there are 16 children more, everyone gets ₹ 10 less. Let the number of children in the orphanage home be \( x \) and the amount to be donated to each child be \( y \).
Based on the above information, answer the following questions:
Rupal, Shanu and Trisha were partners in a firm sharing profits and losses in the ratio of 4:3:1. Their Balance Sheet as at 31st March, 2024 was as follows:
(i) Trisha's share of profit was entirely taken by Shanu.
(ii) Fixed assets were found to be undervalued by Rs 2,40,000.
(iii) Stock was revalued at Rs 2,00,000.
(iv) Goodwill of the firm was valued at Rs 8,00,000 on Trisha's retirement.
(v) The total capital of the new firm was fixed at Rs 16,00,000 which was adjusted according to the new profit sharing ratio of the partners. For this necessary cash was paid off or brought in by the partners as the case may be.
Prepare Revaluation Account and Partners' Capital Accounts.