8. Let 𝑛(𝑥) = 𝑥
2 − 4. The average rate of change of 𝑛(𝑥) over the interval [𝑐, 5] is equal to 3, where 𝑐 is a
constant. Find the value of 𝑐.

Question

8. Let 𝑛(𝑥) = 𝑥
2 − 4. The average rate of change of 𝑛(𝑥) over the interval [𝑐, 5] is equal to 3, where 𝑐 is a
constant. Find the value of 𝑐.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The average rate of change of a function $ n(x) $ over the interval $[c, 5]$ is given by the formula: $ \frac{n(5) - n(c)}{5 - c} = 3 $. ‖ ‖ step 2 ⋮ First, we calculate $ n(5) $: $ n(5) = 5^2 - 4 = 25 - 4 = 21 $. ‖ ‖ step 3 ⋮ Next, we express $ n(c) $: $ n(c) = c^2 - 4 $. ‖ ‖ step 4 ⋮ Substituting into the average rate of change formula gives: $ \frac{21 - (c^2 - 4)}{5 - c} = 3 $. Simplifying this leads to $ \frac{25 - c^2}{5 - c} = 3 $. ‖ ‖ step 5 ⋮ Cross-multiplying results in: $ 25 - c^2 = 3(5 - c) $, which simplifies to $ 25 - c^2 = 15 - 3c $. ‖ ‖ step 6 ⋮ Rearranging gives: $ c^2 - 3c + 10 = 0 $. We can solve this quadratic equation using the quadratic formula: $ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ where $ a = 1, b = -3, c = 10 $. ‖ ‖ step 7 ⋮ Calculating the discriminant: $ (-3)^2 - 4(1)(10) = 9 - 40 = -31 $. Since the discriminant is negative, there are no real solutions for $ c $. ‖ ∻Answer∻ ⚹ There are no real values for $ c $ that satisfy the given conditions. ⚹ ∻Key Concept∻ ⚹ The average rate of change of a function measures how the function's output changes with respect to its input over a specific interval. ⚹ ∻Explanation∻ ⚹ In this case, the average rate of change was set to 3, but the resulting quadratic equation had no real solutions, indicating that no such $ c $ exists in the real number system. ⚹◊What is the formula for calculating the average rate of change of a function over an interval?⍭ Generate me a similar question◊

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