Differentiability flash cards

1. 1.If $f(x)$ is differentiable at $x=a$, what can be concluded about continuity, and does the converse hold?

   Differentiability at $x=a$ $\implies$ continuity at $x=a$. The converse is false — e.g. $f(x)=|x|$ is continuous at $x=0$ but not differentiable there.

   Hint: One-way implication only

2. 2.State the standard test for differentiability of $f$ at $x=a$ using left-hand and right-hand derivatives.

   $f$ is differentiable at $a$ iff $Lf'(a)=\lim_{h\to0^-}\dfrac{f(a+h)-f(a)}{h}$ and $Rf'(a)=\lim_{h\to0^+}\dfrac{f(a+h)-f(a)}{h}$ both exist and are equal.

   Hint: LHD = RHD

3. 3.If $f$ and $g$ are both non-differentiable at $x=a$, is $f+g$ necessarily non-differentiable at $x=a$?

   No. Sum, difference, or product of two non-differentiable functions can be differentiable; e.g. $f(x)=|x|$, $g(x)=-|x|$ gives $f+g=0$, which is differentiable everywhere.

   Hint: Counterexample exists

4. 4.For $f(x)=|x-a|\,\phi(x)$ where $\phi(x)$ is continuous, when is $f$ differentiable at $x=a$?

   $f$ is differentiable at $x=a$ if and only if $\phi(a)=0$ (assuming $\phi$ is continuous at $a$); otherwise the corner from $|x-a|$ persists and $f$ is not differentiable at $a$.

   Hint: Kink vanishes only if $\phi(a)=0$

5. 5.What does Darboux's theorem say about the derivative function $f'$, and why is it significant for JEE problems?

   If $f$ is differentiable on $[a,b]$, then $f'$ takes every value between $f'(a)$ and $f'(b)$ (Intermediate Value Property), even if $f'$ itself is not continuous. This is used to rule out certain functions as derivatives of any function.

   Hint: IVP for derivatives, no continuity needed

6. 6.Examine differentiability of $f(x)=|x|^3$ at $x=0$, and find $f'(0)$ if it exists.

   For $x\ge0$, $f(x)=x^3$; for $x<0$, $f(x)=-x^3$. LHD $=\lim_{h\to0^-}\dfrac{-h^3-0}{h}=\lim_{h\to0^-}(-h^2)=0$. RHD $=\lim_{h\to0^+}\dfrac{h^3-0}{h}=\lim_{h\to0^+}h^2=0$. Since LHD $=$ RHD $=0$, $f$ is differentiable at $0$ with $f'(0)=0$ — the cubic order overrides the modulus.

   Hint: Expand $|x|^3$ separately for $x\ge0,\ x<0$ and compute one-sided derivatives.

7. 7.Show that $f(x)=x|x|$ is differentiable for all real $x$ and find $f'(x)$.

   For $x>0$, $f=x^2$, $f'=2x$; for $x<0$, $f=-x^2$, $f'=-2x$. At $x=0$: LHD $=\lim_{h\to0^-}\dfrac{-h^2}{h}=\lim(-h)=0$; RHD $=\lim_{h\to0^+}\dfrac{h^2}{h}=\lim h=0$. So $f$ is differentiable everywhere with $f'(x)=2|x|$.

   Hint: Split into cases $x>0,x<0$ and check $x=0$ via first principles.

8. 8.Determine all points where $f(x)=|x|+|x-1|$ fails to be differentiable on $\mathbb{R}$.

   $f$ is piecewise linear with slope $-2$ on $(-\infty,0)$, slope $0$ on $(0,1)$, and slope $2$ on $(1,\infty)$. At $x=0$: LHD $=-2\ne0=$ RHD. At $x=1$: LHD $=0\ne2=$ RHD. So $f$ is not differentiable exactly at $x=0$ and $x=1$.

   Hint: Write $f$ as a linear function on each of the three intervals split by $0$ and $1$.

9. 9.Is $f(x)=|\sin x|$ differentiable at $x=\pi$? Compare with its behaviour at $x=\pi/2$.

   Near $\pi$, $\sin x$ changes sign with $\sin'(\pi)=\cos\pi=-1\ne0$. For $x\to\pi^-$, $\sin x>0$ so $f=\sin x$, giving LHD $=\cos\pi=-1$; for $x\to\pi^+$, $\sin x<0$ so $f=-\sin x$, giving RHD $=-\cos\pi=1$. LHD $\ne$ RHD, so $f$ is not differentiable at $\pi$. At $x=\pi/2$, $\sin x=1>0$ locally, so $f=\sin x$ there smoothly, and $f$ is differentiable with $f'(\pi/2)=\cos(\pi/2)=0$.

   Hint: Where does $\sin x$ vanish with nonzero slope, versus stay one-signed?

10. 10.Check differentiability of $f(x)=|x^2-3x+2|$ at $x=1$ and at $x=2$.

    $g(x)=x^2-3x+2=(x-1)(x-2)$ has simple zeros at $1$ and $2$, with $g'(1)=-1\ne0$ and $g'(2)=1\ne0$. At a simple zero of a differentiable $g$ with $g'\ne0$, $|g|$ always has unequal one-sided derivatives $(\mp g'(x_0))$, producing a corner. Hence $f$ is not differentiable at both $x=1$ and $x=2$.

    Hint: Use the rule: $|g(x)|$ is non-differentiable at simple zeros of $g$.

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