Register

GO Classes DPP | GATE CS, DA | Calculus & Optimization | One-Sided Derivatives

187 views
2 2 votes

Let $f:\mathbb{R}\to\mathbb{R}$ be an even function. Suppose the left derivative of $f$ at $x=3$ exists and is equal to $-6$. Which of the following is necessarily true?

  1. The right derivative of $f$ at $x=-3$ need not exist.
     
  2. The right derivative of $f$ at $x=-3$ exists and is equal to $-6$.
     
  3. The right derivative of $f$ at $x=-3$ exists and is equal to $6$.
     
  4. The left derivative of $f$ at $x=-3$ exists and is equal to $6$.

1 Answer

1 1 vote

Since $f$ is even, $f(-x)=f(x)$ for all $x$.

The right derivative at $x=-3$ is

$\lim_{h\to0^+}\frac{f(-3+h)-f(-3)}{h}$

Using evenness,

$f(-3+h)=f(3-h)$ and $f(-3)=f(3)$.

So,

$\lim_{h\to0^+}\frac{f(-3+h)-f(-3)}{h}=\lim_{h\to0^+}\frac{f(3-h)-f(3)}{h}$

Rewrite the denominator as a negative left-side change:

$\lim_{h\to0^+}\frac{f(3-h)-f(3)}{h}=-\lim_{h\to0^+}\frac{f(3-h)-f(3)}{-h}$

But $\lim_{h\to0^+}\frac{f(3-h)-f(3)}{-h}$ is the left derivative of $f$ at $x=3$, which is $-6$.

Therefore, the right derivative at $x=-3$ is $-(-6)=6$.

Answer: C.

Voting & Accepting an answer helps improve the community
Answer:
← Previous · GO Classes GATE CS/D...Next → · GO Classes DPP | GAT...
← Previous in category · GO Classes DPP | GAT...Next in category → · GO Classes DPP | GAT...
Position:
Show:

Related questions

1 1 vote
1 1 answer
248
248 views
GO Classes asked May 29
248 views
GO Classes DPP | GATE CS, DA | Calculus & Optimization | Continuity & Differentiability
The function $$f(x)=\begin{cases}x\sin\left(\frac{1}{x}\right),&x\neq0\\\\0,&x=0\end{cases}$$ at $x=0$ is:Continuous and differentiable Continuous but not differentiable ...
0 0 votes
1 1 answer
260
260 views
GO Classes asked May 29
260 views
GO Classes DPP | GATE CS, DA | Calculus & Optimization | Arc Length
The length of the curve $y=\frac{1}{3}\left((x-1)^2+2\right)^{3/2}$ for $-1\leq x\leq2$ is:$5$ $6$ $\dfrac{20}{3}$ $8$
0 0 votes
1 1 answer
188
188 views
GO Classes asked May 29
188 views
GO Classes DPP | GATE CS, DA | Calculus & Optimization | Area Between Curves
The area of the region between the curves $y=x^2-2x$ and $y=2x-x^2$ for $x\in[-1,3]$ is:$4$ $\frac{16}{3}$ $\frac{20}{3}$ $8$
1 1 vote
1 1 answer
199
199 views
GO Classes asked May 29
199 views
GO Classes DPP | GATE CS, DA | Calculus & Optimization | Inflection Points
The point of inflection of the function $y=e^x(\sin x-\cos x)$ in $[0,\pi]$ is:$x=\dfrac{\pi}{4}$ $x=\dfrac{3\pi}{4}$ $x=\dfrac{\pi}{4},\dfrac{3\pi}{4}$ No point of infle...