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If the function \( f(x)=\left(\dfrac{1}{x}\right)^{2x} \); \( x>0 \) attains the maximum value at \( x=\dfrac{1}{e} \) then:

Suppose for a differentiable function \( h \), \( h(0) = 0 \), \( h(1) \) and \( h'(0) = h'(1) = 2 \). If \( g(x) = h(e^{x})\,e^{h(x)} \), then \( g'(0) \) is equal to:

Let \( f(x) = 4\cos^{3}x + 3\sqrt{3}\cos^{2}x - 10 \). The number of points of local maxima of \( f \) in interval \( (0,\, 2\pi) \) is:

The number of critical points of the function \( f(x) = (x-2)^{2/3} (2x+1) \) is:

Let \( f : \mathbb{R} - \{0\} \to \mathbb{R} \) be a function satisfying \( f\!\left(\dfrac{x}{y}\right) = \dfrac{f(x)}{f(y)} \) for all \( x,y \), \( f(y) \neq 0 \). If \( f'(1)=2024 \), then:

Let \( f:\mathbb{R}\to\mathbb{R} \) be a thrice differentiable function such that \( f(0)=0 \), \( f(1)=1 \), \( f(2)=-1 \), \( f(3)=2 \) and \( f(4)=-2 \). Then, the minimum number of zeros of \(( 3f' f''+ff''')(x) \) is ___________.

If the function \( f(x)=2x^3-9ax^2+12a^2x+1, \ a>0 \) has a local maximum at \( x=\alpha \) and a local minimum at \( x=\alpha^2 \), then \( \alpha \) and \( \alpha^2 \) are the roots of the equation:

Let \( f(x)=2^{x}-x^{2},\ x\in \mathbb{R} \). If \( m \) and \( n \) are respectively the number of points at which the curves \( y=f(x) \) and \( y=f'(x) \) intersect the \( x \)-axis, then the value of \( m+n \) is __________.

Let \(\displaystyle g(x)=3\,f\!\left(\frac{x}{3}\right)+f(3-x) \) and \( f''(x)>0 \) for all \( x\in(0,3) \). If \( g \) is decreasing in \( (0,\alpha) \) and increasing in \( (\alpha,3) \), then \( 8\alpha \) is:

The function \( f(x)=\dfrac{x}{x^{2}-6x-16}, \ x\in \mathbb{R}-\{-2,8\}. \)

In an A.P., the sixth term \( a_6 = 2 \). If the product \( a_1 a_4 a_5 \) is the greatest, then the common difference of the A.P. is equal to:

Let \( S = (-1, \infty) \) and \( f : S \to \mathbb{R} \) be defined as \( f(x) = \int_{-1}^{x} (e^{t} - 1)^{11}(2t - 1)^{5}(t - 3)^{12}(2t - 10)^{61} dt \).
Let \( p \) = Sum of squares of the values of \( x \), where \( f(x) \) attains local maxima on \( S \), and \( q \) = Sum of the values of \( x \), where \( f(x) \) attains local minima on \( S \). Then, the value of \( p^{2} + 2q \) is __________.

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left(1 + x(\lambda^2 - x^2)\right)$ satisfies $\frac{x^2 + x + 2}{x^2 + 5x + 6} < 0$, be $(\alpha, \beta)$. Then $\alpha^2 + \beta^2$ is equal to _______.

For the function \( f(x)=\sin x+3x-\dfrac{2}{\pi}(x^{2}+x) \), where \( x\in\left[0,\dfrac{\pi}{2}\right] \), consider the following two statements:
(\(I\)) \( f \) is increasing in \( \left(0,\dfrac{\pi}{2}\right) \).
(\(II\)) \( f' \) is decreasing in \( \left(0,\dfrac{\pi}{2}\right) \).
Between the above two statements,

Let \( f:\mathbb{R}\to (0,\infty) \) be strictly increasing function such that \( \displaystyle\lim_{x\to\infty}\dfrac{f(7x)}{f(x)}=1 \). Then, the value of \( \displaystyle \lim_{x\to\infty}\left[\dfrac{f(5x)}{f(x)}-1\right] \) is equal to:

The function \( f(x) = 2x + 3(x)^\dfrac{2}{3}, \, x \in R \) has:

The maximum area of a triangle whose one vertex is at \( (0, 0) \) and the other two vertices lie on the curve \( y = -2x^2 + 54 \) at points \( (x, y) \) and \( (-x, y) \), where \( y > 0 \), is:

Let \( f(x) = (x + 3)^2 (x - 2)^3 ,\ x \in [-4, 4] \). If \( M \) and \( m \) are the maximum and minimum values of \( f \), respectively in \( [-4, 4] \), then the value of \( M - m \) is:

If \( 5f(x) + 4f\!\left( \frac{1}{x} \right) = x^2 - 2 \), \( \forall x \neq 0 \) and \( y = 9x^2 f(x) \), then \( y \) is strictly increasing in:

The interval in which the function \( f(x) = x^x, \ x > 0 \), is strictly increasing is:

If range of \( f(\theta) = \dfrac{\sin^4 \theta + 3\cos^2 \theta}{\sin^4 \theta + \cos^2 \theta} , \\ \theta \in \mathbb{R} \) is \( [\alpha, \beta] \), then the sum of the infinite G.P., whose first term is \( 64 \) and the common ratio is \( \dfrac{\alpha}{\beta} \), is equal to ______.

Consider the function \( f : \left[ -\frac{1}{2}, 1 \right] \to \)ℝ defined by \( f(x) = 4\sqrt{2}\,x^3 - 3\sqrt{2}\,x - 1 \). Consider the statements:
(I) The curve \( y = f(x) \) intersects the \( x \)-axis exactly at one point
(II) The curve \( y = f(x) \) intersects the \( x \)-axis at \( x = \cos\frac{\pi}{12} \)
Then:

Let the set of all values of \( p \), for which \( f(x) = (p^2 - 6p + 8)(\sin^2 2x - \cos^2 2x) + 2(2 - p)x + 7 \) does not have any critical point, be the interval \( (a, b) \). Then \( 16ab \) is equal to ________.

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JEE Main 2024 8th April Evening Shift

If the function \( f(x)=2x^3-9ax^2+12a^2x+1, \ a>0 \) has a local maximum at \( x=\alpha \) and a local minimum at \( x=\alpha^2 \), then \( \alpha \) and \( \alpha^2 \) are the roots of the equation: