Applications of Derivatives
Absolute Maxima and Minima
Grade 12
Question:
<p>The absolute minimum value of the function \(\frac{x + 2}{\sqrt{x^2 + 1}}\) is</p>
<p>(a) -1</p>
<p>(b) 1</p>
<p>(c) -5/3</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Evaluate the function behavior at critical points and as x approaches infinity to find the absolute minimum.
<p>Let f(x) = (x + 2)/√(x² + 1)</p><p>f'(x) = [√(x² + 1) - (x + 2) · x/√(x² + 1)] / (x² + 1)</p><p>f'(x) = [(x² + 1) - x(x + 2)] / (x² + 1)^(3/2) = (1 - 2x) / (x² + 1)^(3/2)</p><p>f'(x) = 0 gives x = 1/2</p><p>As x → -∞, f(x) → -1</p><p>f(1/2) = (5/2)/√(5/4) = (5/2)/(√5/2) = √5</p><p>The absolute minimum is -1 (as x → -∞)</p><p>∴ Answer is (a) -1.</p>
Correct Answer: a