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Question
The function f(x) = x |x| is ______.
Options
continuous and differentiable at x = 0
continuous but not differentiable at x = 0
differentiable but not continuous at x = 0
neither differentiable nor continuous at x = 0
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Solution
The function f(x) = x |x| is continuous and differentiable at x = 0.
Explanation:
Given, the function is f(x) = x |x| for x ∈ R.
The function can be written as,
f(x) = `{{:( x^2; x > 0),(-x^2; x ≤ 0):}`
Now, Rf(0) = `lim_(x rightarrow 0^+) (x^2)` = 0
and Lf(0) = `lim_(x rightarrow 0^-) (-x^2)` = 0
So, Lf(0) = Rf(0) = f(0)
So, the function is continuous at 0.
Now, Rf'(0) = `lim_(x rightarrow 0^+) (f(x) - f(0))/(x - 0)`
= `lim_(x rightarrow 0^+) (x^2 - 0)/x` = 0
and Lf'(0) = `lim_(x rightarrow 0^-) (f(x) - f(0))/(x - 0)`
= `lim_(x rightarrow 0^-) (-x^2 - 0)/x` = 0
So, Lf'(0) = Rf'(0)
So, function is differentiable at 0.
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