Question

The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.

Options

• e

• `1/e`

• e²

• `1/e^2`

Answer 1

The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is `underlinebb(1/e)`.

Explanation:

Since, f(x) = `logx/x`

After differentiating on both sides w.r.t.x, we get

f'(x) = `(x. 1/x - log x.1)/x^2 = (1 - log x)/x^2`

For maximum or minimum value of f(x), put f'(x) = 0

`\implies (1 - logx)/x^2` = 0

`\implies` log x = 1

`\implies` x = e

Now, f"(x) = `(3 + 2logx)/x^3`

∴ f"(e) = `- 1/e^3 < 0`

After substituting x = e in equation (i), we get

f(e) = `loge/e = 1/e`

Hence, the maximum value of f(x) is `1/e` at x = e.