Question 19 of 82beginner🔧 ApplyShort Answer3 marks

Show that the function given by f(x) = 3x + 17 is increasing on R.

Correct Answer

Given f(x) = 3x + 17

Differentiating with respect to x: f'(x) = 3

Since f'(x) = 3 > 0 for all x ∈ R, the function f is increasing on R.

Alternatively, let x₁ and x₂ be any two numbers in R such that x₁ < x₂. Then, x₁ < x₂ ⇒ 3x₁ < 3x₂ ⇒ 3x₁ + 17 < 3x₂ + 17 ⇒ f(x₁) < f(x₂)

Thus, by definition, f is strictly increasing on R.

Exercise: EXERCISE 6.2 | Q: 1 | (Page 12)
For More Understanding

Explanation

This question tests understanding of increasing functions. The context provides two methods: (1) Using the derivative test from Theorem 1 - since f'(x) = 3 > 0 for all x ∈ R, the function is increasing; (2) Using the definition approach as shown in Example 7 for a similar function f(x) = 7x - 3. Both methods are valid for proving that a linear function with positive slope is increasing on R.

Solution Steps

  1. Step 1: Find the derivative of f(x) = 3x + 17, which gives f'(x) = 3.

  2. Step 2: Observe that f'(x) = 3 > 0 for all x ∈ R.

  3. Step 3: Apply Theorem 1 - since f'(x) > 0 for each x ∈ R, the function f is increasing on R.

PreviousNext