Advertisements
Advertisements
Question
At what point on the following curve, is the tangent parallel to x-axis y = x2 on [−2, 2]
?
Advertisements
Solution
Let \[f\left( x \right) = x^2\]
Since \[f\left( x \right)\] is a polynomial function, it is continuous on \[\left[ - 2, 2 \right]\] and differentiable on \[\left( - 2, 2 \right)\] .
Consequently, there exists at least one point c
APPEARS IN
RELATED QUESTIONS
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
f (x) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
f (x) = x2/3 on [−1, 1] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = (x − 1) (x − 2)2 on [1, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 4)2 on the interval [0, 4] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x2 + 5x + 6 on the interval [−3, −2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex cos x on [−π/2, π/2] ?
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) = sin x − sin 2x on [0, π]?
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] ?
If f : [−5, 5] → R is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x2 − 2x + 4 on [1, 5] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = 2x − x2 on [0, 1] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore f(x) = (x − 1)(x − 2)(x − 3) on [0, 4] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore f(x) = tan−1 x on [0, 1] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem \[f\left( x \right) = x + \frac{1}{x} \text { on }[1, 3]\] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem \[f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]\] ?
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?
Let C be a curve defined parametrically as \[x = a \cos^3 \theta, y = a \sin^3 \theta, 0 \leq \theta \leq \frac{\pi}{2}\] . Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).
If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)n on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?
Find the value of c prescribed by Lagrange's mean value theorem for the function \[f\left( x \right) = \sqrt{x^2 - 4}\] defined on [2, 3] ?
If the polynomial equation \[a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0\] n positive integer, has two different real roots α and β, then between α and β, the equation \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].
If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]
The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is
When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x is ______.
The value of c in Rolle's theorem for the function f (x) = x3 − 3x in the interval [0,\[\sqrt{3}\]] is
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
The maximum value of sinx + cosx is ______.
At what point, the slope of the curve y = – x3 + 3x2 + 9x – 27 is maximum? Also find the maximum slope.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
The minimum value of `1/x log x` in the interval `[2, oo]` is
