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प्रश्न
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
उत्तर
We may rewrite f as f(x) = `{{:(x^2",", "if" x ≥ 0),(-x^2",", "if" x < 0):}`
Now Lf ′(0) = `lim_("h" -> 0^-) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^-) (-"h"^2 - 0)/"h"`
= `lim_("h" -> 0^-) - "h"`
= 0
Now Rf ′(0) = `lim_("h" -> 0^+) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^+) ("h"^2 - 0)/"h"`
= `lim_("h" -> 0^+) "h"`
= 0
Since the left hand derivative and right hand derivative both are equal, hence f is differentiable at x = 0.
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