Practicing Success
For x > 1 and y = log x which one of the following is not true ? |
$x-1>y$ $x^2-1>y$ $y>x-1$ $\frac{x-1}{x}<y$ |
$y>x-1$ |
Let $f(x)=\log x-(x-1)$. Then, $f'(x)=\frac{1}{x}-1=\frac{1-x}{x}$ Clearly, f'(x) < 0 for x > 1 ⇒ f(x) is decreasing function for x > 1 ⇒ f(x) < f(1) for x > 1 ⇒ log x - (x - 1) < 0 for x > 1 ⇒ log x < x - 1 for x > 1 But, $x^2-1>x-1$ for x > 1 ∴ $x^2-1>x-1$ and $\log x<x-1 \Rightarrow \log x<x^2-1$ Similarly, it can be proved that $\frac{x-1}{x}<\log x$. Hence, (a), (b) and (d) are true. Consequently, option (c) is not true. |