If \(f(x) = \cos\left\{\dfrac{\pi}{3}[x] + x\right\}\) for \(1 < x < 2\), where \([\cdot]\) is the greatest integer function, then what is \(f\left(\dfrac{\pi}{2}\right)\) equal to?
\(-\dfrac{\sqrt{3}}{2}\)
Since \(\dfrac{\pi}{2}\) lies between \(1\) and \(2\), \(\left[\dfrac{\pi}{2}\right] = 1\), so \(f(x) = \cos\left(\dfrac{\pi}{3} + x\right)\) on this interval. Thus \(f\left(\dfrac{\pi}{2}\right) = \cos\left(\dfrac{\pi}{3} + \dfrac{\pi}{2}\right) = \cos\left(\dfrac{5\pi}{6}\right) = -\dfrac{\sqrt{3}}{2}\).
Similar Questions
Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?
The value of x, satisfying the equation \(log_{cos x} ~sin x = 1\) , where \(0<x<\dfrac{\pi}{2}\) , is
If \({{x}^{{{\log }_{7}}x}}>7\) where x > 0, then which one of the following is correct?
If f(x) = log 10 (1 + x), then what is 4f(4) + 5f(1) – log 10 2 equal to?
A function f defined by f(x) = In \(\left( {\sqrt {{x^2} + 1} - x} \right)\) is
If f(x) = 3 1+x , then f(x) f(y) f(z) is equal to
If (0.2) x= 2 and log 10 2 = 0.3010, the what is the value of x to the nearest tenth?
If n = (2017)! then what is \(\frac{1}{{{{\log }_2}n}} + \frac{1}{{{{\log }_3}n}} + \frac{1}{{{{\log }_4}n}} + \ldots + \frac{1}{{{{\log }_{2017}}n}}\) equal to?
IF x + log 15 (1 + 3 x) = x log 15 5 + log 15 12, where x is an integer, then what is x equal to?
What is \(\frac{1}{{{{\log }_2}N}} + \frac{1}{{{{\log }_3}N}} + \frac{1}{{{{\log }_4}N}} + \ldots + \frac{1}{{{{\log }_{100}}N\;}}\;\) equal to (N ≠ 1)?
Important Questions from Special Functions
- The function $f(x) = [2x]$ where $[x]$ is the greatest integer function, is continuous at
The number of real solutions of equation x 2 - 3 |x| + 2 = 0 is:
If ϕ is the Euler’s Totient function, then ϕ(92) is:
Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongruent solutions modulo 9 of this congruence are:
If log10(x2 - 6x + 45) = 2, then the value of x are:
