If f(x) = | cases x & x^ 2 & x^ 3 1 & 2x & 3x^ 2 0 & 2 & 6x cases | , then the ratio f^ '' ( x ) : f^ ' ( x ) =

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Differentiation

JEE Main · Mathematics · Differentiation

If f(x) = | cases x & x^ 2 & x^ 3 1 & 2x & 3x^ 2 0 & 2 & 6x cases |...

f\left(x\right) = \left|

\begin{cases}x & x^{2} & x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 2 & 6x\end{cases}

\right|

, then the ratio

f^{''} ( x ) : f^{'} ( x ) =

Options

A $2: x$
B $x^{2}: x$
C $3x: 2$
D $6: x$

Correct answer

A. $2: x$

Step-by-step solution

if& $f\left(x\right) =f\left(x\right) = \left|$ $\begin{array}{ll}x x^{2} x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 2 & 6x\end{array}$ $\right|$

$f^{'}\left(x\right) = \left|$ \begin{cases}1 & 2x & 3x^{2} \\ 1 & 2x & 3x^{2} \\ 0 & 2 & 6x\end{cases} $\right| + \left|$ \begin{cases}x & x^{2} & x^{3} \\ 0 & 2 & 6x \\ 0 & 2 & 6x\end{cases} $\right| + \left|$ \begin{cases}x & x^{2} & x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 0 & 6\end{cases} $\right|$

$= 0 + 0 + \left|$ \begin{cases}x & x^{2} & x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 0 & 6\end{cases} $\right|$

$= 6x^{2}$

$f^{''}\left(x\right) = 12x$

$f^{''} ( x ) : f^{'} ( x )$

$= 12x: 6x^{2}$

$= 2: x$

Practice Differentiation on Quantrex Academy →

More from Differentiation

Let ℝ denote the set of all real numbers. Consider the polynomial function f: ℝ arrow ℝ… JEE Advanced 2026 Let f(x) and g(x) be twice differentiable functions satisfying f''(x) = g''(x) for all x… JEE Main 2026 Let f be a real polynomial of degree n such that f(x) = f'(x) f''(x) , for all x ∈ ℝ . If… JEE Main 2026 Let f(x)=x^ 3 +x^ 2 f^ (1)+2 x f^ (2)+f^ (3), x ∈ R . Then the value of f^ (5) is: JEE Main 2026 Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ and g: ℝ arrow(0,4) be… JEE Advanced 2025 Let f: R arrow R be a twice differentiable function such that ( x y)(f(2 x+2 y)-f(2 x-2… JEE Main 2025 If _e y=3 ^ -1 x , then (1-x^2) y^ -x y^ at x=(1)/(2) is equal to JEE Main 2024 Let f(x)=a x^3+b x^2+c x+41 be such that f(1)=40, f^ (1)=2 and f^ (1)=4 . Then… JEE Main 2024 Suppose for a differentiable function h, h(0)=0, h(1)=1 and h^ (0)=h^ (1)=2 . If… JEE Main 2024 If f(x)=\ array l x^3 ((1)/(x)), x ≠ 0 0 , x=0 array . then JEE Main 2024 If y(θ)=(2 θ+ 2 θ)/( 3 θ+4 2 θ+5 θ+2) , then at θ=(π)/(2), y^ +y^ +y is equal to : JEE Main 2024 Let f(x)=x^5+2 e^ x / 4 for all x ∈ R . Consider a function g(x) such that (g ∘ f)(x)=x… JEE Main 2024 If y = ( x + 1)( x ^ 2 - x ) x x + x + x + (1)/(15)(3 ^ 2 x - 5) ^ 3 x, then 96 y ^ '… JEE Main 2024 Let y = _ e ( 1 - x ^ 2 1 + x ^ 2 ), - 1 < x < 1 . Then at x = (1)/(2) , the value of… JEE Main 2024 Suppose f(x) = ( 2 ^ x + 2 ^ - x ) x ^ - 1 ( x ^ 2 - x + 1) (7 x ^ 2 + 3x + 1) ^ 3 . Then… JEE Main 2024 Let f ( x ) = x ^ 3 + x ^ 2 f ^ ' ( 1 ) + x f ^ " ( 2 ) + f ^ ''' ( 3 ) , x ∈ R . Then f… JEE Main 2024 Full Differentiation list Differentiation JEE Maths PYQs JEE Mathematics PYQs All 99 Percentile Qs Bank for JEE Main PYQs