List of top Mathematics Questions on Maxima and Minima

The value of x is maximum for
Let \( f(x) = x^2 + ax + \beta \). If \( f \) has a local minimum at \( (2,6) \), then \( f(0) \) is equal to
The maximum value of the function \( f(x) = -2x^2 + 4x + 1 \) occurs at:
The absolute maximum value of $y = x^{3} - 3x + 2$, for $0 \leq x \leq 2$, is:
Let \( \alpha_1 \) and \( \beta_1 \) be the distinct roots of \( 2x^2 + (\cos\theta)x - 1 = 0, \ \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha_1 + \beta_1 \), then \( 16(M + m) \) equals:
The slope of the curve \( y = -x^3 + 3x^2 + 8x - 20 \) is maximum at:
If \( f(x) = \lfloor x \rfloor \) is the greatest integer function, then the correct statement is:
If surface area \( S \) is constant, the volume \( V = \frac{1}{4}(Sx - 2x^3) \), \( x \) being the edge of the base. Show that the volume \( V \) is maximum for \( x = \frac{\sqrt{6}}{6} \).
Let $ f(x) = |x^2 - 4x + 3| + |x^2 - 5x + 6| $. The minimum value of $ f(x) $ is:
If the expression \(x + \frac{1}{x^2},\; x>0\) attains minimum value at \(x=\alpha\), then \(\alpha^3\) is:
Subject to constraints: 2x + 4y ≤ 8, 3x + y ≤ 6, x + y ≤ 4, x, y ≥ 0; The maximum value of Z = 3x + 15y is:
Let \( f(x) = x^3 - 6x^2 + 12x - 3 \), then at \( x = 2 \), \( f(x) \) has:
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:
If \( x \) is real and \( \alpha, \beta \) are maximum and minimum values of \( \frac{x^2 - x + 1}{x^2 + x + 1} \) respectively, then \( \alpha + \beta = \):
Let \( f(x) = 3\sqrt{x - 2} + \sqrt{4 - x} \) be a real-valued function. If \( \alpha \) and \( \beta \) are respectively the minimum and the maximum values of \( f \), then \( \alpha^2 + 2\beta^2 \) is equal to:
Let the maximum and minimum values of \[\left( \sqrt{8x - x^2 - 12 - 4} \right)^2 + (x - 7)^2, \quad x \in \mathbb{R} \text{ be } M \text{ and } m \text{ respectively}.\] Then \( M^2 - m^2 \) 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 the set of all positive values of \( \lambda \), for which the point of local minimum of the function \((1 + x (\lambda^2 - x^2)) \frac{x^2 + x + 2}{x^2 + 5x + 6} < 0\) be \((\alpha, \beta)\).
Then \( \alpha^2 + \beta^2 \) is equal to ________.
What rebate in price of the calculator should the store give to maximise the revenue?
Determine the number of units (\( x \)) that should be sold to maximise the revenue \( R(x) = x \cdot p(x) \). Also verify the result.

Define \( f(x) = \begin{cases} x^2 + bx + c, & x< 1 \\ x, & x \geq 1 \end{cases} \). If f(x) is differentiable at x=1, then b−c is equal to

If \( M_1 \) and \( M_2 \) are the maximum values of \( \frac{1}{11 \cos 2x + 60 \sin 2x + 69} \) and \( 3 \cos^2 5x + 4\sin^2 5x \) respectively, then \( \frac{M_1}{M_2} = \):
The minimum value of the function \( x^2 + y^2 + z^2 \) if \( x + y + z = 3a \)

For the function \(f(x) = x^2 e^{-x},\) the maximum occurs when \( x \) is equal to ____ .