Question

number of values of x belons to [0,pi] where f(x)=[4sinx-7] is not derivable is?

Answer 1

9

Answer 2

Let

$$g(x) = 4\sin x - 7.$$

The function $f(x) = [\,g(x)\,]$ (floor function) is not differentiable at the points where $g(x)$ is an integer. For $x \in [0,\pi]$, since $\sin x \in [0,1]$, we have:

$$g(x) \in [4\cdot0-7,\,4\cdot1-7] = [-7,-3].$$

Thus, the possible integer values are:

$$k = -7,\,-6,\,-5,\,-4,\,-3.$$

For each integer $k$, we solve:

$$4\sin x - 7 = k \quad \Rightarrow \quad \sin x = \frac{k+7}{4}.$$

• For $k = -7$:
  $\sin x = 0$.
  Solutions in $[0,\pi]$ are $x = 0$ and $x = \pi$ (2 values).

• For $k = -6$:
  $\sin x = \frac{1}{4}$.
  Two solutions (one in $(0, \pi/2)$ and one in $(\pi/2, \pi)$) (2 values).

• For $k = -5$:
  $\sin x = \frac{1}{2}$.
  Solutions: $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$ (2 values).

• For $k = -4$:
  $\sin x = \frac{3}{4}$.
  Two solutions in $[0,\pi]$ (2 values).

• For $k = -3$:
  $\sin x = 1$.
  The unique solution is $x = \frac{\pi}{2}$ (1 value).

Total points of non-differentiability:

$$2+2+2+2+1=9.$$