The set of critical points of the function f(x)=x-logx+∫2^π(1 / z-2-2 cos4 z) d z is (A) {(π)/(6

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The set of critical points of the function...

The set of critical points of the function $f(x)=x-\log x+\int_{2}^{\pi}(1 / z-2-2 \cos 4 z) d z$ is (A) $\left\{\frac{\pi}{6}, \frac{n \pi}{2}+\frac{\pi}{6}\right\}, n \in N$ (B) $\{n \pi\}, n \in N$ (C) $\left\{\frac{\pi}{2}, \mathrm{n} \pi+\frac{\pi}{6}\right\}, \mathrm{n} \in \mathrm{N}$ (D) None of these

The set of critical points of the function $f(x)=x-\log x+\int_{2}^{\pi}(1 / z-2-2 \cos 4 z) d z$ is
(A) $\left\{\frac{\pi}{6}, \frac{n \pi}{2}+\frac{\pi}{6}\right\}, n \in N$
(B) $\{n \pi\}, n \in N$
(C) $\left\{\frac{\pi}{2}, \mathrm{n} \pi+\frac{\pi}{6}\right\}, \mathrm{n} \in \mathrm{N}$
(D) None of these

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Key Concepts

Critical Points

Critical points of a function are those points in its domain where the derivative is either zero or undefined. These points are important because they often indicate local minima, local maxima, or points of inflection in the function.

Derivative

The derivative of a function represents its instantaneous rate of change with respect to its variable. Finding where the derivative equals zero is a key method in locating potential extrema, as these are the points where the function's behavior may change.

Constants in Differentiation

When differentiating a function, any additive constant—including those coming from definite integrals with fixed limits—disappears, meaning it does not affect the calculation of the derivative or the location of the critical points.

Logarithmic Differentiation

The process of differentiating logarithmic functions, such as the natural logarithm, involves using specific rules (for example, the derivative of log x is 1/x). This concept is essential when the function includes logarithmic terms, as it directly impacts the overall derivative and thus the determination of critical points.

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