If x = √(a^(^(-1)t)) and y = √(a^(^(-1)t)), then d^2y/dx^2 is Option 1: -y/x Option 2: y/x^2 Option 3: 2y/x^2 Option 4: 0

Answer: 2y/x^2. Solution: xy = a^(1/2(^(-1)t+^(-1)t)) = a^(/4), a constant k. So y = k/x, giving dy/dx = -k/x^2 and d^2y/dx^2 = 2k/x^3 = 2y/x^2.

Calculus

Continuity & Differentiability

Hard

core

If $x = \sqrt{a^{\tan^{-1}t}}$ and $y = \sqrt{a^{\cot^{-1}t}}$ , then $\frac{d^2y}{dx^2}$ is

$-\frac{y}{x}$

$\frac{y}{x^2}$

$\frac{2y}{x^2}$

✅ Correct Option: 3

$xy = a^{\frac{1}{2}(\tan^{-1}t+\cot^{-1}t)} = a^{\pi/4}$ , a constant $k$ .

So $y = \frac{k}{x}$ , giving $\frac{dy}{dx} = -\frac{k}{x^2}$ and $\frac{d^2y}{dx^2} = \frac{2k}{x^3} = \frac{2y}{x^2}$ .

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If $\frac{d}{dx}[ax^3 + ax^2 + ax + 1] = 9x^2 + 6x + 3$ , then $a$ is equal to

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The derivative of $(\log x)^x$ with respect to $\log x$ is

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Consider the following statements for the curve $f(x) = x|x|$ and find that which of the following(s) are/is correct :

(A) $f(x) = x|x|$ is differentiable at $x = 0$ .

(B) $f(x) = x|x|$ is continuous at $x = 0$ but not differentiable at $x = 0$ .

(D) $f(x) = x|x|$ is symmetrical about y-axis.

Choose the correct answer from the options given below :

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If $f(x) = |x| + |x - 5|$ , then which of the following statements are TRUE?

(A) f is a continuous function every where

(B) f is a continuous function except $x = 5$ and $x = 0$

(D) f is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$

Choose the correct answer from the options given below:

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Let $y = m\sin rx + n\cos rx$ . What is the value of $\frac{d^2y}{dx^2}$ ?