Advertisements
Advertisements
Question
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?
Advertisements
Solution
We have
\[f\left( x \right) = A x^2 + Bx + C\]
Differentiating the given function with respect to x, we get
\[f'\left( x \right) = 2Ax + B\]
\[\because f\left( a \right) = f\left( b \right)\]
\[ \therefore A a^2 + Ba + C = A b^2 + bB + C\]
\[ \Rightarrow A a^2 + Ba = A b^2 + bB\]
\[ \Rightarrow A\left( a^2 - b^2 \right) + B\left( a - b \right) = 0\]
\[ \Rightarrow A\left( a - b \right)\left( a + b \right) + B\left( a - b \right) = 0\]
\[ \Rightarrow \left( a - b \right)\left[ A\left( a + b \right) + B \right] = 0\]
\[ \Rightarrow a = b, A = \frac{- B}{\left( a + b \right)}\]
\[ \Rightarrow \left( a + b \right) = \frac{- B}{A} \left( \because a \neq b \right)\]
From (1), we have
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) = 3 + (x − 2)2/3 on [1, 3] 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) = cos 2x on [−π/4, π/4] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex sin x 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 \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
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 = \[e^{1 - x^2}\] on [−1, 1] ?
If f : [−5, 5] → R is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?
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 ?
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(x) = x2 + x − 1 on [0, 4] ?
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 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) ?
Find a point on the curve y = x3 + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?
State Lagrange's mean value theorem ?
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] ?
The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
Show that the local maximum value of `x + 1/x` is less than local minimum value.
An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when θ = `pi/6`
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
The minimum value of `1/x log x` in the interval `[2, oo]` is
Let y = `f(x)` be the equation of a curve. Then the equation of tangent at (xo, yo) is :-
What does continuity on a closed interval guarantee?
For a continuous function on the closed interval \[a,d\], what does \[f(a)\] represent?
For a continuous function on the closed interval \[a,d\], what does \[f(d)\] represent?
Which statement expresses the Extreme Value Theorem?
If a differentiable function has an absolute max or min at an interior point \[c\], what must be true?
After finding critical points and identifying endpoints, what should be done next?
For \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\], which derivative is correct?
What is \[f\left(\frac{1}{8}\right)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
