First slide

Differentiability

Question

$Let the function f (x) = \{\begin{array}{cc} \frac{\sin^{3} x^{2}}{x} & \forall x \neq 0 \\ 0 & \forall x = 0 \end{array}$

Moderate

A

is continuous but not derivable at x=0
2)
3)
B

Neither continuous nor differentiable at x = 0
C

continuous but not differentiable at x = 1
D

Continuous and differentiable at x = 0

Solution

$\underset{x \to 0}{L t} f (x) = \underset{x \to 0}{L t} \frac{\sin x^{2}}{x^{2}} (x \sin^{2} x^{2}) = \underset{x \to 0}{L t} (\frac{\sin x^{2}}{x^{2}}) . \underset{x \to 0}{L t} (x \sin^{2} x^{2})$
$= 0 = f (0)$
$And L.H.D at x = 0 is {Lt}_{h \to 0} \frac{f (0 + h) - 0}{h} = \frac{\frac{\sin^{3} h^{2}}{h} - 0}{h} = 0$
R.H.D = 0
$∴ f is differentiable at x = 0$

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