PYQs Based Test - 05 : Maxima, Minima and Mean value theorem

The maximum value of f(x) = 2x³ – 9x² + 12x – 3 in the interval 0 ≤ x ≤ 3 is _______

Let f(x) = x² - 2x + 2 be a continuous function defined on x ∈ [1, 3]. The point x at which the tangent of f(x) becomes parallel to the straight line joining f(1) and f(3) is

Let the function f≤ft( θ ) = ≤ft| {*20c θ &θ &θ \\ ≤ft( π /6 )&≤ft( π /6 )&≤ft( π /6 )\\ ≤ft( π /3 )&≤ft( π /3 )&≤ft( π /3 ) } |

Where $\theta \in \left[\frac{\pi }{6},\frac{\pi }{3}\right]$ and f’(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?

(I) There exists $\theta \in \left(\frac{\pi }{6},\frac{\pi }{3}\right)$ such that f’(θ) = 0

(II) There exists $\theta \in \left(\frac{\pi }{6},\frac{\pi }{3}\right)$ such that f’(θ) ≠ 0

For 0 ≤ t < ∞, the maximum value of the function f(t) = e${}^{-t\text{–}2e}$${}^{-2t}$ occurs at:

In the open interval (0, 1), the polynomial p(x) = x⁴ - 4x³ + 2 has