Applications of Derivatives
Monotonicity and Increasing/Decreasing Functions
Grade 12
Question:
<p>For the function <span>\(y = f(x) = (x - b)x(x - c)e^x\)</span>, which of the following holds?</p>
<p>(a) If <span>\(f(x) > 0\)</span> for all real <span>\(x\)</span> ⟹ <span>\(f'(x) > 0\)</span></p>
<p>(b) If <span>\(f(x) > 0\)</span> for all real <span>\(x\)</span> ⟹ <span>\(f'(x) > 0\)</span></p>
<p>(c) If <span>\(f'(x) > 0\)</span> for all real <span>\(x\)</span> ⟹ <span>\(f(x) > 0\)</span></p>
<p>(d) If <span>\(f'(x) > 0\)</span> for all real <span>\(x\)</span> ⟹ <span>\(f(x) > 0\)</span></p>
Step-by-Step Solution
Key Concept: Understand the relationship between the sign of the derivative and monotonicity of the function.
<p>Consider the polynomial <span>$y = (x - b)x(x - c)e^x$</span> with roots at <span>$x = 0, b, c$</span>.</p><p>If <span>$f'(x) > 0$</span> for all real <span>$x$</span>, then <span>$f$</span> is strictly increasing everywhere.</p><p>A strictly increasing function can have at most one real root. Since <span>$f(x) = 0$</span> has three real roots, <span>$f$</span> cannot be strictly increasing if <span>$f(x)$</span> changes sign.</p><p>If <span>$f'(x) > 0$</span> for all <span>$x$</span>, then <span>$f$</span> is strictly increasing, hence either always positive or always negative. Given the structure, if it's always positive, this is consistent.</p>
Correct Answer: C