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 \( [1, 2] \).
A compound (A) with molecular formula $C_4H_9I$ which is a primary alkyl halide, reacts with alcoholic KOH to give compound (B). Compound (B) reacts with HI to give (C) which is an isomer of (A). When (A) reacts with Na metal in the presence of dry ether, it gives a compound (D), C8H18, which is different from the compound formed when n-butyl iodide reacts with sodium. Write the structures of A, (B), (C) and (D) when (A) reacts with alcoholic KOH.
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)