JEE MATH APEX

Applications of Derivatives-JEE-Mains-PYQ’s

Mathematics Questions

Consider the following three statements for the function f : (0, \infty) \rightarrow \mathbb{R} defined by f(x) = |\log_e x| - |x - 1| :
(I) f is differentiable at all x > 0.
(II) f is increasing in (0, 1).
(III) f is decreasing in (1, \infty).
Then.
(A) Only (I) is TRUE.
(B) Only (I) and (III) are TRUE.
(C) Only (II) and (III) are TRUE.
(D) All (I), (II) and (III) are TRUE.
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The least value of \left(\cos^2 \theta - 6 \sin \theta \cos \theta + 3 \sin^2 \theta + 2\right) is
(A) 4 - \sqrt{10}
(B) -1
(C) 4 + \sqrt{10}
(D) 1
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Let \alpha and \beta respectively be the maximum and the minimum values of the function
f(\theta) = 4 \left(\sin^4 \left( \frac{7\pi}{2} - \theta \right) + \sin^4(11\pi + \theta)\right) - 2 \left(\sin^6 \left( \frac{3\pi}{2} - \theta \right) + \sin^6(9\pi - \theta)\right), \theta \in \mathbf{R}.
Then \alpha + 2\beta is equal to :
(A) 6
(B) 5
(C) 4
(D) 3
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Let f(x) = x^{2025} - x^{2000}, x \in [0, 1] and the minimum value of the function f(x) in the interval [0, 1] be (80)^{80}(n)^{-81}. Then n is equal to
(A) -40
(B) -41
(C) -80
(D) -81
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Let f : \mathbb{R} \rightarrow \mathbb{R} be a twice differentiable function such that f''(x) > 0 for all x \in \mathbb{R} and f'(a - 1) = 0, where a is a real number.
Let g(x) = f(\tan^2 x - 2 \tan x + a), 0 < x < \frac{\pi}{2}.
Consider the following two statements:
(I) g is increasing in \left(0, \frac{\pi}{4}\right)
(II) g is decreasing in \left(\frac{\pi}{4}, \frac{\pi}{2}\right)
Then,
(A) Both (I) and (II) are True
(B) Neither (I) nor (II) is True
(C) Only (I) is True
(D) Only (II) is True
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Let f: \mathbb{R} \rightarrow \mathbb{R} be a polynomial function of degree four having extreme values at x = 4 and x = 5. If \lim_{x \to 0} \frac{f(x)}{x^2} = 5, then f(2) is equal to:
(A) 8
(B) 10
(C) 12
(D) 14
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Let the function \( f(x) = \frac{x}{3} + \frac{3}{x} + 3, x \neq 0 \) be strictly increasing in \( (-\infty, \alpha_1) \cup (\alpha_2, \infty) \) and strictly decreasing in \( (\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5) \). Then \( \sum_{i=1}^{5} \alpha_i^2 \) is equal to
(A) 48
(B) 40
(C) 36
(D) 28
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Let \( x = -1 \) and \( x = 2 \) be the critical points of the function \( f(x) = x^3 + ax^2 + b \log_e |x| + 1, x \neq 0 \). Let \( m \) and \( M \) respectively be the absolute minimum and the absolute maximum values of \( f \) in the interval \( \left[-2, -\frac{1}{2}\right] \). Then \( |M + m| \) is equal to (Take \( \log_e 2 = 0.7 \)):
(A) 21.1
(B) 19.8
(C) 22.1
(D) 20.9
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Let \( a > 0 \). If the function \( f(x) = 6x^3 – 45ax^2 + 108a^2x + 1 \) attains its local maximum and minimum values at the points \( x_1 \) and \( x_2 \) respectively such that \( x_1 x_2 = 54 \), then \( a + x_1 + x_2 \) is equal to:
(A) 15
(B) 13
(C) 24
(D) 18
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Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = ||x + 2| – 2|| \). If \( m \) is the number of points of local minima and \( n \) is the number of points of local maxima of \( f \), then \( m + n \) is
(A) 3
(B) 3
(C) 2
(D) 5
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