For the function $f(x)=(x-1)(x-2)$ defined on $\left[0, \frac{1}{2}\right]$,the value of $c$ satisfying Lagrange's mean value theorem is

Let $f(x)$ be a differentiable function for all $x$ such that $f'(x) \le 5$ and $f(1) = 4$. The maximum value of $f(5)$ is...

The equation $2^x+5^x=3^x+4^x$ has

Let $f(x)$ be a function continuous on $[1, 2]$ and differentiable on $(1, 2)$ satisfying $f(1) = 2, f(2) = 3$ and $f'(x) \geq 1$ for all $x \in (1, 2)$. Define $g(x) = \int_1^x f(t) \, dt$ for all $x \in [1, 2]$. Then the greatest value of $g(x)$ on $[1, 2]$ is-

If $f(x)=(2x-1)(3x+2)(4x-3)$ is a real-valued function defined on $[\frac{1}{2}, \frac{3}{4}]$, then the value$(s)$ of '$c$' as defined in the statement of Rolle's theorem is/are:

Let $f(x) = 8x^3 - 6x^2 - 2x + 1,$ then