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

MCQ

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

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

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