Which of the following functions satisfies all conditions of the Rolle’s theorem in the intervals specified?

Question

Which of the following functions satisfies all conditions of the Rolle’s theorem in the intervals specified? (A) f(x)=x(1/3),x∈ [- 2,3] (B) f(x)=sin x,x∈ [- π , (π /6)] (C) f(x)=ln ((x2 + a b/x (a + b))), x∈ [a , b],0 < a < b (D) f(x)=ex2 - x,x∈ [0,4]

Tardigrade Admin · Accepted Answer

f(x)=x(1/3) is not differentiable at x=0 For f(x)=sin x,f(- π )≠ f((π /6)) For f(x)=ln ((x2 + a b/x (a + b))) f(a)=f(b)=ln (1)=0 Also, it is continuous in [a , b] and differentiable in (a , b) . Hence, Rolle&rsquo;s Theorem is applicable. For f(x)=ex2 - x,f(0)≠ f(4)

Q. Which of the following functions satisfies all conditions of the Rolle’s theorem in the intervals specified?

$f (x) = x^{\frac{1}{3}}, x \in [- 2, 3]$

$f (x) = s in x, x \in [- π, \frac{π}{6}]$

$f (x) = l n (\frac{x ^{2} + ab}{x ( a + b )}),$ $x \in [a, b], 0 < a < b$

$f (x) = e^{x^{2} - x}, x \in [0, 4]$

Q. Which of the following functions satisfies all conditions of the Rolle’s theorem in the intervals specified?

f(x)=x31​,x∈[−2,3]

f(x)=sinx,x∈[−π,6π​]

f(x)=ln(x(a+b)x2+ab​), x∈[a,b],0<a<b

f(x)=ex2−x,x∈[0,4]

f(x)=x31​ is not differentiable at x=0 For f(x)=sinx,f(−π)=f(6π​) For f(x)=ln(x(a+b)x2+ab​) f(a)=f(b)=ln(1)=0 Also, it is continuous in [a,b] and differentiable in (a,b) . Hence, Rolle’s Theorem is applicable. For f(x)=ex2−x,f(0)=f(4)

$f (x) = x^{\frac{1}{3}}, x \in [- 2, 3]$

$f (x) = s in x, x \in [- π, \frac{π}{6}]$

$f (x) = l n (\frac{x ^{2} + ab}{x ( a + b )}),$ $x \in [a, b], 0 < a < b$

$f (x) = e^{x^{2} - x}, x \in [0, 4]$