Let f(x) and g(x) be two functions which are defined and differentiable for all x \geq x_0 . If f(x_0) = g(x_0) and f'(x) > g'(x) for all x > x_0 , then

Step 1: Consider h(x) = f(x) - g(x) Step 2: Then h(x_0) = f(x_0) - g(x_0) = 0 Step 3: And h'(x) = f'(x) - g'(x) > 0 for all x > x_0 Step 4: Since h'(x) > 0 , the function h(x) is strictly increasing for x > x_0 Step 5: Therefore, h(x) > h(x_0) = 0 for all x > x_0 Step 6: This means f(x) > g(x) for all x > x_0 ∴ Answer is (d).

Applications of Derivatives

Monotonicity and Inequalities

Grade 12

Question:

Let f(x) and g(x) be two functions which are defined and differentiable for all x \geq x_0. If f(x_0) = g(x_0) and f'(x) > g'(x) for all x > x_0, then

(a) f(x) > g(x) for some x > x_0

(b) f(x) = g(x) for some x > x_0

(c) f(x) > g(x) only for some x > x_0

(d) f(x) > g(x) for all x > x_0

Step-by-Step Solution

Key Concept: If two functions start equal and one has a consistently larger derivative, it will be larger for all subsequent values.

Step 1: Consider h(x) = f(x) - g(x)Step 2: Then h(x_0) = f(x_0) - g(x_0) = 0Step 3: And h'(x) = f'(x) - g'(x) > 0 for all x > x_0Step 4: Since h'(x) > 0, the function h(x) is strictly increasing for x > x_0Step 5: Therefore, h(x) > h(x_0) = 0 for all x > x_0Step 6: This means f(x) > g(x) for all x > x_0∴ Answer is (d).

Correct Answer: D

Question:

Step-by-Step Solution

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