F x - k -1-1-2 cos2 x -3 k -3- then the number of points where - xf x - xf x - x -1- x 2-3 x -2- is non-differentiable in x -0-3 - is equal to where - denotes the greatest integer function

F(x) =∏_k=1^∞ nbsp;(1 + 2 cos(2x/3^k)/3) = ∏_k=1^∞1/3(1 + 2 4 sin^2 x/3^k) =∏_k=1^∞1/3 sin(x/3^k){3 sin(x/3^k) 4 sin^3 (x/3^k)} =∏_k=1^∞ nbsp;[sin(x/3^k ...

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Byju's Answer

Standard XII

Mathematics

Factorization Method Form to Remove Indeterminate Form

f x =∏ k =1∞1...

Question

$f (x) = \infty \prod k = 1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{1 + 2 cos (\frac{2 x}{3^{k}})}{3} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠$ , then the number of points where $[x f (x)] + | x f (x) | + (x - 1) | x^{2} - 3 x + 2 |$ is non-differentiable in $x \in (0, 3 π)$ is equal to (where [.] denotes the greatest integer function)

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Solution

$f (x) = \infty \prod k = 1 ⎛ ⎜ ⎝ \frac{1 + 2 cos (\frac{2 x}{3^{k}})}{3} ⎞ ⎟ ⎠ = \infty \prod k = 1 \frac{1}{3} (1 + 2 - 4 {sin}^{2} \frac{x}{3^{k}}) = \infty \prod k = 1 \frac{1}{3 sin (\frac{x}{3^{k}})} {3 sin (\frac{x}{3^{k}}) - 4 {sin}^{3} (\frac{x}{3^{k}})} = \infty \prod k = 1 ⎡ ⎢ ⎣ \frac{sin (\frac{x}{3^{k - 1}})}{3 sin (\frac{x}{3^{k}})} ⎤ ⎥ ⎦ = lim k \to \infty \frac{1}{3^{k}} ⎧ ⎪ ⎨ ⎪ ⎩ \frac{sin x}{sin \frac{x}{3}} \times \frac{sin \frac{x}{3}}{sin \frac{x}{9}} \times . . . . . \times \frac{sin (\frac{x}{3^{k - 1}})}{sin (\frac{x}{3^{k}})} ⎫ ⎪ ⎬ ⎪ ⎭ = lim k \to \infty \frac{sin x}{\frac{x sin (\frac{x}{3^{k}})}{(\frac{x}{3^{k}})}} \Rightarrow f (x) = \frac{sin x}{x} x f (x) = sin x$
So $[sin x] + | sin x | + (x - 1) | (x - 1) (x - 2) |$
is non-differentiable at $5 (\frac{π}{2}, π, 2 π, \frac{5 π}{2}, 2)$ points

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f (x) = x | x^{2} - 3 x | e^{| x - 3 |} + [\frac{3 + s g n (x^{2} - 1 - 5 x)}{4 + x^{2}}] + {\frac{1}{3} + [\frac{x^{2} - 2 x}{| x | + 1}]}

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f(x)=∞∏k=1⎛⎜ ⎜ ⎜ ⎜⎝1+2cos(2x3k)3⎞⎟ ⎟ ⎟ ⎟⎠, then the number of points where [xf(x)]+|xf(x)|+(x−1)|x2−3x+2| is non-differentiable in x∈(0,3π) is equal to (where [.] denotes the greatest integer function)