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CUET Mathematics 2024

Calculus

>

Continuity & Differentiability

Medium

If $\mathrm{e}^{\mathrm{y}}=\mathrm{x}^{\mathrm{x}}$ , then which of the following is true ?

y \frac{d^{2} y}{d x^{2}}=1

\frac{d^{2} y}{d^{2}}-y=0

\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0

y \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+1=0

Correct Option: 4

We are given:

e^y = x^x

Take natural log on both sides

y = \ln(x^x) = x \ln x

First derivative:

\frac{dy}{dx} = \frac{d}{dx}(x \ln x) = \ln x + 1

Second derivative:

\frac{d^2y}{dx^2} = \frac{d}{dx}(\ln x + 1) = \frac{1}{x}

\Rightarrow \frac{dy}{dx} = \ln x + 1

\Rightarrow \frac{d^2y}{dx^2} = \frac{1}{x}

\Rightarrow y = x \ln x

Let’s check option (4):

y \cdot \frac{d^2y}{dx^2} - \frac{dy}{dx} + 1

= (x \ln x)\left(\frac{1}{x}\right) - (\ln x + 1) + 1

= \ln x - \ln x - 1 + 1 = 0

Sahi hai, aur kya chahiye? Option 4 is correct!

Related questions:

CUET Mathematics 2024

f(x)

f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.

is continuous at

x=\pi

, then the value of

k

CUET Mathematics 2024

[x]

denote the greatest integer function. Then match List-I with List-II:

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List-I	List-II
(A) $\|x - 1\| + \|x - 2\|$	(I) is differentiable everywhere except at $x = 0$
(B) $x - \|x\|$	(II) is continuous everywhere
(C) $x - [x]$	(III) is not differentiable at $x = 1$
(D) $x \, \|x\|$	(IV) is differentiable at $x = 1$

CUET Mathematics 2024

The second order derivative of which of the following functions is

5^{\mathrm{x}}