If f \left(\right. x \left.\right) = \left{\right. - x - \frac{\pi}{2} , x \leq \frac{- \pi}{2} \\ - cosx , \frac{- \pi}{2} < x \leq 0 \\ x - 1 , 0

Engineering

Mathematics

Important Points on Derivability

Question

If $f (x) = {\begin{cases} - x - \frac{π}{2}, x \leq \frac{- π}{2} \\ - cosx, \frac{- π}{2} < x \leq 0 \\ x - 1, 0 < x \leq 1 \\ \ln x, x > 1 \end{cases}$

then

f (x) is not differentiable at x = 0

f (x) is continuous at $x = \frac{- π}{2}$

f (x) is differentiable at x = 1

f (x) is differentiable at $x = \frac{- 3}{2}$

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$At x = - \frac{π}{2}$

LHL = 0, RHL = 0, $f (- \frac{π}{2}) = 0$

So f(x) is continuous at $x = - \frac{π}{2}$

At x = 0

LHD = 0 ; RHD = 1

So f(x) is not differentiable at x = 0

At x = 0

LHD = 1, RHD = 1

So f(x) is differentiable at x = 1

$in (- \frac{π}{2}, 0); f (x) = - \cos x$

So f(x) is differentiable at $x = - \frac{3}{2}$