Differentiating inequality

Question

So I was solving a question that goes as follows:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and satisfy $$ \left| f(x) \right| \leq x^4 + 5x^2\qquad \forall x $$ Show that $f'(0) = 0$

T.P.T means to prove that

Now I know we can't differentiate the function due to inequality. After some time when I was not able to solve the question I looked up the solution on Youtube. The teacher there solved the question using a first principles method .

My Doubt

Isn't first principle same as differentiating a function . (Because this is how we prove the standard formulae ex F'(x) of sinx = cosx )

If we cant differentiate the function then how can we use first principle to solve this question?
Can this question be solved in any other way ?

Check out the squeeze theorem, noting that $f(0) = 0$ by the inequality. — Cameron Williams, Oct 31 '19 at 12:13
@dfnu https://brilliant.org/wiki/derivative-by-first-principle/ — B Luthra, Oct 31 '19 at 17:02

score 1 · Answer 1 · answered Oct 31 '19 at 12:15

1

$|f(0)| \leq 0$ which means $f(0)=0$. Hence $|\frac {f(x)-f(0)} x| \leq |x^{3}+5x| \to 0$ as $ x \to 0$. Hence $f'(0)=0$ by definition of $f'(0)$.

answered Oct 31 '19 at 12:15

Kavi Rama Murthy

311,013

but isn't this same as differentiating f(x) as we are using first principle here ? – B Luthra Oct 31 '19 at 12:25

Differentiating inequality

1 Answers1