Let f(x) = x^3 -6x^2 +15x + 3. Then: - Vidyadip

MCQ

Let $f(x) = x^3 -6x^2 +15x + 3.$ Then:

A
$f(x) > 0$ for all $\text{x}\in\text{R}.$
B
$f(x) > 0$ for all $\text{x}\in\text{R}.$
✓
$f(x)$ is invertible.
D
None of these.

✓

Answer

Correct option: C.

$f(x)$ is invertible.

$f(x) = x^3 - 6x^2 +15x + 3$
$f'(x) = 3x^2 - 12x + 15$
$= 3(x^2 - 4x + 5)$
$= 3(x^2 - 4x + 4 + 1)$
$=3(\text{x}-2)^2+\frac{1}{3}>0$
Therefore, $f(x)$ is strictly increasing function.
$\Rightarrow f^{-1}(x)$ exists.
Hence, $f(x)$ is an invertible function.

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