#### Question

Let f(x) = x^{3} − 6x^{2} + 15x + 3. Then,

(a) f(x) > 0 for all x ∈ R

(b) f(x) > f(x + 1) for all x ∈ R

(c) f(x) is invertible

(d) none of these

#### Solution

(c) f(x) is invertible

f(x) =x^{3} − 6x^{2} + 15x + 3

\[f'(x) = 3 x^2 - 12x + 15\]

\[ = 3\left( x^2 - 4x + 5 \right)\]

\[ = 3\left( x^2 - 4x + 4 + 1 \right)\]

\[ = 3 \left( x - 2 \right)^2 + \frac{1}{3} > 0\]

\[\text { Therefore, f(x) is strictly increasing function }. \]

\[ \Rightarrow f^{- 1} (x) \text { exists } . \]

\[\text { Hence, f(x) is an invertible function } .\]

Is there an error in this question or solution?

Solution Let F(X) = X3 − 6x2 + 15x + 3. Then, (A) F(X) > 0 For All X ∈ R (B) F(X) > F(X + 1) for All X ∈ R (C) F(X) is Invertible (D) None of These Concept: Increasing and Decreasing Functions.