Let f(x) be a function satisfying |f(x)| ≤ x^2 for -1 ≤ x ≤ 1, how do you show that f is differentiable at x = 0 and find f’(0)?

Answer 1

The derivative exists and is zero.

Since #|f(x)|<=x^2# for #xin [-1,1]#, we must have #|f(0)| <= 0^2=0#, but since it is definitely non-negative, it must be 0.
#f(0) = 0#

Now

#f^'(0) = lim_{h to 0} (f(0+h)-f(0))/h = lim_(h to 0) f(h)/h#

Thus

#|f^'(0)| = lim_{h to 0}|f(h)/h|= lim_{h to 0} |f(h)|/h # #qquad <= lim_(h to 0) h^2/h=lim_(h to 0) h = 0#
Hence #|f^'(0)| <= 0 implies |f^'(0)| = 0#
Hence the derivative at #x=0# exists and is 0.
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Answer 2

To show that ( f(x) ) is differentiable at ( x = 0 ), we need to demonstrate that the limit ( \lim_{{h \to 0}} \frac{f(0 + h) - f(0)}{h} ) exists.

Given ( |f(x)| \leq x^2 ) for ( -1 \leq x \leq 1 ), we have ( -x^2 \leq f(x) \leq x^2 ). Since ( f(0) = 0 ), it follows that ( -x^2 \leq f(x) \leq x^2 ) for ( -1 \leq x \leq 1 ).

Now, considering ( \lim_{{h \to 0}} \frac{f(0 + h) - f(0)}{h} ), we have:

( \lim_{{h \to 0}} \frac{f(h)}{h} \leq \lim_{{h \to 0}} \frac{h^2}{h} = \lim_{{h \to 0}} h = 0 )

And

( \lim_{{h \to 0}} \frac{f(-h)}{h} \geq \lim_{{h \to 0}} \frac{-h^2}{h} = \lim_{{h \to 0}} -h = 0 )

Thus, both the left-hand and right-hand limits are equal to zero, which means ( f'(0) = 0 ).

Therefore, ( f(x) ) is differentiable at ( x = 0 ) and ( f'(0) = 0 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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