<p>If \(y = e^{ax^{-4x}}\) and \(z = e^{-\ cos^{-1}x}\) then \(\dfrac{d^2y}{dz^2} = 0\).</p><p><strong>State whether the statement is true or false.</strong></p>
Step-by-Step Solution
Key Concept: To verify if d²y/dz² = 0, we must compute dy/dz and d²y/dz² using the chain rule and quotient rule systematically. The statement is FALSE because d²y/dz² ≠ 0 for the given functions.
<p><strong>Step 1:</strong> Identify the functions: y = e^(ax² - 4x) and z = e^(-cos⁻¹x)</p><p><strong>Step 2:</strong> Find dy/dx and dz/dx:</p><p>dy/dx = e^(ax² - 4x) · (2ax - 4)</p><p>dz/dx = e^(-cos⁻¹x) · (-1/√(1-x²)) · (-1) = e^(-cos⁻¹x) / √(1-x²)</p><p><strong>Step 3:</strong> Find dy/dz using the chain rule:</p><p>dy/dz = (dy/dx)/(dz/dx) = [e^(ax² - 4x)(2ax - 4)] / [e^(-cos⁻¹x) / √(1-x²)]</p><p>dy/dz = e^(ax² - 4x + cos⁻¹x) · (2ax - 4) · √(1-x²)</p><p><strong>Step 4:</strong> Find d²y/dz² = d/dz(dy/dz) = [d/dx(dy/dz)] / (dz/dx)</p><p>The numerator d/dx(dy/dz) is a product of exponential and polynomial terms that does NOT simplify to zero.</p><p><strong>Step 5:</strong> Since d/dx(dy/dz) ≠ 0, we have d²y/dz² ≠ 0</p><p>∴ Answer: <strong>FALSE</strong></p>
Correct Answer: A