Definite Integration
Properties and Applications of Definite Integrals
Grade 12
Question:
<p>Let <i>f(x)</i> be a polynomial function of degree 3 where <i>a < b < c</i> and <i>f(a) = f(b) = f(c)</i>. If the graph of <i>f(x)</i> is as shown, which of the following statements are INCORRECT? (Where <i>c > |a|</i>)</p><p>(a) \(\int_a^c f(x)dx = \int_b^c f(x)dx + \int_a^b f(x)dx\)</p><p>(b) \(\int_a^c f(x)dx < 0\)</p><p>(c) \(\int_a^b f(x)dx < \int_c^b f(x)dx\)</p><p>(d) \(\frac{1}{b-a}\int_a^b f(x)dx > \frac{1}{c-b}\int_b^c f(x)dx\)</p>
<p>(a) \(\int_a^c f(x)dx = \int_b^c f(x)dx + \int_a^b f(x)dx\)</p>
<p>(b) \(\int_a^c f(x)dx < 0\)</p>
<p>(c) \(\int_a^b f(x)dx < \int_c^b f(x)dx\)</p>
<p>(d) \(\frac{1}{b-a}\int_a^b f(x)dx > \frac{1}{c-b}\int_b^c f(x)dx\)</p>
Step-by-Step Solution
Key Concept: Use properties of definite integrals and analyze areas above/below the x-axis from the given graph of a cubic polynomial
<p><strong>Analysis:</strong> Given a cubic polynomial with three equal values at <i>a, b, c</i>, we analyze each statement:</p><p>(a) By additive property of integrals: $\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$. This is CORRECT.</p><p>(b) From the graph, the negative area below the x-axis is larger than the positive area above, so $\int_a^c f(x)dx < 0$. This is CORRECT, making it INCORRECT to state.</p><p>(c) $\int_a^b f(x)dx$ is positive (above x-axis) while $\int_c^b f(x)dx = -\int_b^c f(x)dx$ is positive (negative of negative area). Comparing magnitudes shows the statement is INCORRECT.</p><p>(d) The mean value of <i>f</i> on <i>[a,b]</i> is greater than on <i>[b,c]</i> in magnitude but opposite sign makes this INCORRECT.</p><p>∴ Answer is (b, c, d)</p>
Correct Answer: b, c, d