Let \(g(x) = \int_0^x f(t) dt\) , where f is such that \(1 \leq f(t) \leq 2\) for \(t \in [0,1]\) and \(0 \leq f(t) \leq 1/2\) for \(t \in (1,2]\) . Then \(g(2)\) satisfies the inequality

Definite Integration

Integral Bounds and Inequalities

Grade 12

Question:

Let \(g(x) = \int_0^x f(t) dt\), where f is such that \(1 \leq f(t) \leq 2\) for \(t \in [0,1]\) and \(0 \leq f(t) \leq 1/2\) for \(t \in (1,2]\). Then \(g(2)\) satisfies the inequality

(A) \(-3/2 \leq g(2) < 1/2\)

(B) \(0 \leq g(2) < 2\)

(C) \(3/2 < g(2) \leq 5/2\)

(D) \(2 < g(2) < 4\)

Step-by-Step Solution

Key Concept: Use the bounds on f(t) to establish bounds on g(2) by splitting the integral into two parts.

IIT - 2000

Correct Answer: C

Question:

Step-by-Step Solution

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