Question

On which of the following functions can we apply LMVT in the interval [-2,2] ?

• A. Y=|x|
• B. y = e^x
• C. y=x³
• D. y=[x]
• E. y=cosx

Answer 1

Answer: (B), (C), (E)

The correct options are
B

y = e^x

C

y=x³

E

y=cosx

LMVT says that if y = f(x) be a given function which is ;

1.Continuous in [a,b]

2.Differentiable in (a,b)

Then, f'(c) = $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ for some c $ϵ$ (a , b)

So, to apply LMVT the function must be continuous and differentiable in the given interval

A.We know that |x| has a sharp edge at x = 0 making it non-differentiable at x = 0. So we can’t apply LMVT on |x|

B. y = $e^x$is continuous and differentiable in its domain. So we can apply LMVT on this function

C. $x^3$ This is also one of the answers

D. Y= =[x]. As we know, [x] is discontinuous at all the integers. So we can’t apply LMVT

E. Y= = cosx is continuous and differentiable at all the points. Hence we can apply LMVT on cosx

So the answers are B, C and E