If \( f(x) = x^3 - x^2 f'(1) + x f''(2) - f'''(3), \, x \in \mathbb{R} \), then:
\( f(3) - f(2) = f(1) \)
\( 2f(0) - f(1) + f(3) = f(2) \)
\( f(1) + f(2) + f(3) = f(0) \)
\( 3f(1) + f(2) = f(3) \)
$\lim_{t \to 0} \left( 1^{\frac{1}{\sin^2 t}} + 2^{\frac{1}{\sin^2 t}} + \dots + n^{\frac{1}{\sin^2 t}} \right)^{\sin^2 t} \text{ is equal to:}$
If \( f(x) = x^2 + g'(1)x + g''(2) \) and \( g(x) = f(1)x^2 + x f'(x) + f''(x) \), then the value of \( f(4) - g(4) \) is equal to ________.
Let \( y(x) = (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 + x^{16}) \). Then \( y' - y'' \) at \( x = -1 \) is equal to:
If \(\displaystyle \lim_{x \to 0} \dfrac{e^{ax} - \cos(bx) - \dfrac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17\), then \(5a^2+b^2\) is equal to:
Let \( f(x)=\displaystyle\sum_{k=1}^{10}k\,x^{k},\; x\in \mathbb{R} \). If \( 2f(2)+f'(2)=119(2)^{n}+1 \) then \( n \) is equal to __________.
Let \( f \), \( g \) and \( h \) be the real valued functions defined on \( \mathbb{R} \) as:and \( h(x)=2[x]-f(x) \), where \( [x] \) is the greatest integer \( \leq x \).Then the value of \( \lim\limits_{x\to 1} g\!\left(h(x-1)\right) \) is:
Let \( [ x] \) denote the greatest integer function and \( f(x)=\max\{\,1+x+[ x ], \ 2+x, \ x+2[ x ]\,\} \), \( 0\le x\le 2 \). Let \( m \) be the number of points in \([0,2]\), where \( f \) is not continuous and \( n \) be the number of points in \( (0,2) \), where \( f \) is not differentiable. Then \( (m+n)^{2}+2 \) is equal to:
Let \( f(x)=[x^2-x]+|-x+[x]| \), where \( x\in \mathbb{R} \) and \( [t] \) denotes the greatest integer less than or equal to \( t \). Then, \( f \) is:
If \( 2x^{y} + 3y^{x} = 20 \), then \( \dfrac{dy}{dx} \) at \( (2, 2) \) is equal to:
Let \( k \) and \( m \) be positive real numbers such that the function is differentiable for all \(x>0\).Then\(\displaystyle \frac{8 f'(8)}{f'\left(\dfrac{1}{8}\right)}\) is equal to ______.
For the differentiable function \( f:\mathbb{R}-\{0\}\to \mathbb{R} \), let \( 3f(x)+2f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10 \), then \( \left|\,f(3)+f'\!\left(\dfrac{1}{4}\right)\,\right| \) is equal to:
Let \( [ x ] \) be the greatest integer \( \leq x \). Then the number of points in the interval \( (-2,1) \), where the function \( f(x)=| [ x ] |+\sqrt{x-[ x ]} \) is discontinuous, is_____.
Then at \( x=0 \)
The value of \( \displaystyle \lim_{n \to \infty} \dfrac{1+2-3+4+5-6+\ldots+(3n-2)+(3n-1)-3n}{\sqrt{2n^{4}+4n+3}-\sqrt{n^{4}+5n+4}} \) is:
Let \( f(x) = \dfrac{\sin x + \cos x - \sqrt{2}}{\sin x - \cos x},\; x \in [0, \pi] - \left\{ \dfrac{\pi}{4} \right\}. \) Then \( f\!\left(\dfrac{7\pi}{12}\right) f''\!\left(\dfrac{7\pi}{12}\right) \) is equal to:
is differentiable for all \(x>0\).Then\(\displaystyle \frac{8 f'(8)}{f'\left(\dfrac{1}{8}\right)}\) is equal to ______.