On which of the following intervals is the function f given by \(f(x)=x^{100}+sin\ x-1\) strictly decreasing?
\((0,1)\)
\((\frac \pi2,\pi)\)
\((0,\frac \pi2)\)
\(None\ of \ these\)
We have,
f(x) = x100+sinx-1
f'(x) = 100x99+cosx
In interval(0,1), cosx>0 and 100x99>0.
f'(x)>0.
Thus, function f is strictly increasing in interval (0, 1).
In interval (\(\frac \pi2\),\(\pi\)), cosx<0 and 100x99>0. also, 100x100>cosx
\(\implies\)f'(x)>0 in (\(\frac \pi2\),\(\pi\)).
Thus, function f is strictly increasing in interval (\(\frac \pi2\),\(\pi\)).
In interval (0,\(\frac \pi2\)), cosx>0 and 100x99>0.
100x99+cosx>0
f'(x)>0 on (0,\(\frac \pi2\)).
Hence, function f is strictly decreasing in none of the intervals. The correct answer is (D).
If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \]
Find the least value of ‘a’ for which the function \( f(x) = x^2 + ax + 1 \) is increasing on the interval \( [1, 2] \).
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.
Increasing Function:
On an interval I, a function f(x) is said to be increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≤ f(y)
Decreasing Function:
On an interval I, a function f(x) is said to be decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≥ f(y)
Strictly Increasing Function:
On an interval I, a function f(x) is said to be strictly increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) < f(y)
Strictly Decreasing Function:
On an interval I, a function f(x) is said to be strictly decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) > f(y)