Question

activity - comparing non-linear functions

activity
value: 1

which function has more number of zeros?
a. \\(y = 3x - 4\\)
b. \\(y = x^3 + 4x - 9\\)
c. \\(y = x^5 - 5x^3\\)
d. \\(y = 3x^3 - 4x^4\\)

Explanation:

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<reasoning>

Analyze the degree of each polynomial function

The maximum number of real zeros a polynomial function can have is equal to its degree.

• Option a: \(y = 3x - 4\) (Degree 1, maximum 1 zero)
• Option b: \(y = x^3 + 4x - 9\) (Degree 3, maximum 3 zeros)
• Option c: \(y = x^5 - 5x^3\) (Degree 5, maximum 5 zeros)
• Option d: \(y = 3x^3 - 4x^4\) (Degree 4, maximum 4 zeros)

Determine the actual number of real zeros for each function

• Option a: \(3x - 4 = 0 \implies x = \frac{4}{3}\) (Exactly 1 real zero)
• Option b: \(y = x^3 + 4x - 9\). Since \(y' = 3x^2 + 4 > 0\), the function is strictly increasing. It has exactly 1 real zero.
• Option c: \(x^5 - 5x^3 = 0 \implies x^3(x^2 - 5) = 0 \implies x = 0, \pm\sqrt{5}\) (Exactly 3 distinct real zeros)
• Option d: \(3x^3 - 4x^4 = 0 \implies x^3(3 - 4x) = 0 \implies x = 0, \frac{3}{4}\) (Exactly 2 distinct real zeros)

Compare the number of zeros

Comparing the number of distinct real zeros:

• Option a: 1 zero
• Option b: 1 zero
• Option c: 3 zeros
• Option d: 2 zeros

Thus, the function with the most zeros is \(y = x^5 - 5x^3\).
</reasoning>

<answer>
<mcq-option>a. \(y = 3x - 4\)</mcq-option>
<mcq-option>b. \(y = x^3 + 4x - 9\)</mcq-option>
<mcq-correct>c. \(y = x^5 - 5x^3\)</mcq-correct>
<mcq-option>d. \(y = 3x^3 - 4x^4\)</mcq-option>
</answer>

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Answer:

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Analyze the degree of each polynomial function

The maximum number of real zeros a polynomial function can have is equal to its degree.

• Option a: \(y = 3x - 4\) (Degree 1, maximum 1 zero)
• Option b: \(y = x^3 + 4x - 9\) (Degree 3, maximum 3 zeros)
• Option c: \(y = x^5 - 5x^3\) (Degree 5, maximum 5 zeros)
• Option d: \(y = 3x^3 - 4x^4\) (Degree 4, maximum 4 zeros)

Determine the actual number of real zeros for each function

• Option a: \(3x - 4 = 0 \implies x = \frac{4}{3}\) (Exactly 1 real zero)
• Option b: \(y = x^3 + 4x - 9\). Since \(y' = 3x^2 + 4 > 0\), the function is strictly increasing. It has exactly 1 real zero.
• Option c: \(x^5 - 5x^3 = 0 \implies x^3(x^2 - 5) = 0 \implies x = 0, \pm\sqrt{5}\) (Exactly 3 distinct real zeros)
• Option d: \(3x^3 - 4x^4 = 0 \implies x^3(3 - 4x) = 0 \implies x = 0, \frac{3}{4}\) (Exactly 2 distinct real zeros)

Compare the number of zeros

Comparing the number of distinct real zeros:

• Option a: 1 zero
• Option b: 1 zero
• Option c: 3 zeros
• Option d: 2 zeros

Thus, the function with the most zeros is \(y = x^5 - 5x^3\).
</reasoning>

<answer>
<mcq-option>a. \(y = 3x - 4\)</mcq-option>
<mcq-option>b. \(y = x^3 + 4x - 9\)</mcq-option>
<mcq-correct>c. \(y = x^5 - 5x^3\)</mcq-correct>
<mcq-option>d. \(y = 3x^3 - 4x^4\)</mcq-option>
</answer>

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