Let f(x) = ax2 + bx + c, where a, b, c \in R and a \neq 0 be a quadratic polynomial such that ln(1+ f(x)) \geq 0 \forall x \in R and |f(1) – 4 | + |f(2)| = 0. Identify which of the following statement(s) is(are) correct?

Engineering

Mathematics

Basic Rule of Differentiation

Question

Let f(x) = ax² + bx + c, where a, b, c $\in$ R and a $\neq$ 0 be a quadratic polynomial such that ln(1+ f(x)) $\geq$ 0 $\forall$ x $\in$ R and |f(1) – 4 | + |f(2)| = 0. Identify which of the following statement(s) is(are) correct?

${\frac{d}{dx} (f (f (x))) |}_{x = 1} = 128$

${\frac{d}{dx} (f (f (x))) |}_{x = 2} = 0$

4a + 2b + 5c = 16

4a + 2b + 5c = 64

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ln(1 + f(x)) $\geq$ 0 ⇒ 1 + f(x) $\geq$ 1 ⇒ f (x) $\geq$ 0 $\forall x \in$ R
|f(1) – 4 | + |f(2)| = 0 ⇒ f (1) = 4, f (2) = 0

$∴$ f (x) = a (x – 2)²
f (1) = a · (1) = 4   ⇒ a = 4
⇒ f (x) = 4 (x – 2)²    ⇒ f '(x) = 8 (x – 2)
Now,   4a + 2b + 5c = f (2) + 4 f(0) = 0 + 4 × 16 = 64 ⇒ (A)
$\frac{d}{dx} (f (f (x))) = f' (f (x)) f' (x)$
${\frac{d}{dx} (f (f (x))) |}_{x = 2} = f' (f (2)) f' (2) = 0 \Rightarrow (C)$
${\frac{d}{dx} (f (f (x))) |}_{x = 1} = f' (f (1))$ f'(1) = f '(4) f '(1) = 16 × (–8) = – 128.