Let $ f(x) = cos 2 \pi x + x -[x]$ ([.] denotes the greatest integer function). Then the number of points in [0, 10], at which f (x) assumes its local maximum value, is

10

The correct answer is option (1) : 10

We have,

$f(x) cos 2\pi x + x - [x]\, ∀\, x \in [0, 10]$

$⇒f'(x) = -2 \pi sin 2 \pi x + 1 \, ∀ x \in \underset{k  = 0}{\overset{9}{∪}}(k, k+1)$

$∴f'(x) = 0 ⇒sin 2 \pi x =\frac{1}{2\pi }$

Clearly, this equation has two solutions in each of the sub-intervals $(k, k+1)$, where k = 0, 1, 2, ..., 0.

Since points of maxima and minima occur alternately.

Therefore, out of 20 points, there are 10 points of local maxima and 10 points of local minima.