Question

Does the function f(x)= ln (x+6) on (-5,10) satisfy the hypotheses of the mean value theorem?

Answer 1

No, since. the function attains 0 at -5. According to M.V.T.

$$ \huge \ let f: [a,b] \rightarrow \ R \ be \ continuous \ function \ on \ \\ \huge [a,b] \ and \ differentiable \ on \ (a,b). \\ \huge \ Then \ there \ exist \ some \ c \ in \ (a,b) \ such \ that \\ \huge f'(c) = \dfrac{f(b)-f(a)} { b-a} $$

$$\huge \ Since \ here \ c(-5) \ doesn't \ lie \ in \ (a,b) \\ \huge \ therefore \ M.V.T. \ is \ not \ satisfied.$$