PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Verify Rolle's theorem for the following function on the indicated interval  f(x) = cos 2x on [0, π] ?

Verify Rolle's theorem for the following function on the indicated interval f (x) = \[\frac{\sin x}{e^x}\] on 0 ≤ x ≤ π ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 3x on [0, π] ?

Verify Rolle's theorem for the following function on the indicated interval f (x) = \[{e^{1 - x}}^2\] on [−1, 1] ?

Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x² + 2) − log 3 on [−1, 1] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = 2 sin x + sin 2x on [0, π] ?

Verify Rolle's theorem for the following function on the indicated interval \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?

Verify Rolle's theorem for the following function on the indicated interval \[f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]\] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = 4^sin x on [0, π] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = x² − 5x + 4 on [1, 4] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin⁴ x + cos⁴ x on \[\left[ 0, \frac{\pi}{2} \right]\] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?

Using Rolle's theorem, find points on the curve y = 16 − x², x ∈ [−1, 1], where tangent is parallel to x-axis.

At what point  on the following curve, is the tangent parallel to x-axis y = x² on [−2, 2]
?

At what point  on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?

At what point  on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?

It is given that the Rolle's theorem holds for the function f(x) = x³ + bx² + cx, x  \[\in\] at the point x = \[\frac{4}{3}\] , Find the values of b and c ?

Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?