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

If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?

According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value ξ in this interval such that $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=$

Let f(x) be a real -valued function such that f'(x₀) = 0 for some x₀ ∈ (0, 1), and f${}^{"}$ (x) > 0 for all x ∈ (0, 1). Then f(x) has

A function f(x) = 1 - x² + x³ is defined in the closed interval [-1, 1]. The value of x, in the open interval (-1, 1) for which the mean value theorem is satisfied, is

Consider a differentiable function f(x) on the set of real numbers such that f(−1) = 0 and |f′(x)| ≤ 2. Given these conditions, which one of the following inequalities is necessarily true for all x ∈ [−2 , 2] ?