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# Solution for 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 - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

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

#### Solution

(c) f(x) is invertible
f(x) =x3 − 6x2 + 15x + 3

$f'(x) = 3 x^2 - 12x + 15$

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

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

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

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

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

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

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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.
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