Limits, Continuity & Differentiability
Second Derivatives and Differential Equations
Grade 12
Question:
<p>If <i>f</i>''(<i>x</i>) = <i>f</i>(<i>x</i>), where <i>f</i>(<i>x</i>) is a continuous double differentiable function and <i>g</i>(<i>x</i>) = <i>f</i>'(<i>x</i>). If <i>F</i>(<i>x</i>) = \[8f\left(\frac{x}{2}\right) + 8g\left(\frac{x}{2}\right)\] and <i>F</i>(5) = 5, then <i>F</i>(10) is</p>
<p>(a) 0</p>
<p>(b) 5</p>
<p>(c) 10</p>
<p>(d) 25</p>
Step-by-Step Solution
Key Concept: Use the differential equation f''(x) = f(x) to express f and g in exponential form, then evaluate F at the given points.
<p>Given: <i>f</i>''(<i>x</i>) = <i>f</i>(<i>x</i>), so <i>f</i>(<i>x</i>) = <i>Ae</i><sup><i>x</i></sup> + <i>Be</i><sup>-<i>x</i></sup></p><p><i>g</i>(<i>x</i>) = <i>f</i>'(<i>x</i>) = <i>Ae</i><sup><i>x</i></sup> - <i>Be</i><sup>-<i>x</i></sup></p><p><i>F</i>(<i>x</i>) = 8<i>f</i>(\frac{<i>x</i>}{2}) + 8<i>g</i>(\frac{<i>x</i>}{2})</p><p>= 8<i>Ae</i><sup><i>x</i>/2</sup> + 8<i>Be</i><sup>-<i>x</i>/2</sup> + 8<i>Ae</i><sup><i>x</i>/2</sup> - 8<i>Be</i><sup>-<i>x</i>/2</sup></p><p>= 16<i>Ae</i><sup><i>x</i>/2</sup></p><p>From <i>F</i>(5) = 5: 16<i>Ae</i><sup>5/2</sup> = 5, so <i>Ae</i><sup>5/2</sup> = 5/16</p><p><i>F</i>(10) = 16<i>Ae</i><sup>5</sup> = 16(<i>Ae</i><sup>5/2</sup>)<sup>2</sup> = 16(5/16)<sup>2</sup> = 16 · 25/256 = 25</p>
Correct Answer: D