Applications of Derivatives
Tangent to Curves
Grade 12
Question:
<p>The coordinate of the point(s) on the graph of the function <span class='math'>\(f(x) = \frac{x^3}{3} - \frac{5x^2}{2} + 7x - 4\)</span> where the tangent drawn cuts-off intercepts from the coordinate axes which are equal in magnitude but opposite in sign, is</p>
<p>(a) <span class='math'>\(\left(2, \frac{8}{3}\right)\)</span></p>
<p>(b) <span class='math'>\(\left(3, \frac{7}{2}\right)\)</span></p>
<p>(c) <span class='math'>\(\left(1, \frac{5}{6}\right)\)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: When a tangent cuts equal and opposite intercepts on coordinate axes, its slope equals 1. Find points where dy/dx = 1.
<p><strong>Step 1:</strong> Since intercepts are equal in magnitude but opposite in sign, the tangent line passes through point (x, y) on the curve and makes equal intercepts of opposite sign on axes.</p><p><strong>Step 2:</strong> For a tangent with slope m at point (x, y), the condition for equal and opposite intercepts is <span class='math'>$\frac{dy}{dx} = 1$</span></p><p><strong>Step 3:</strong> Calculate derivative: <span class='math'>$\frac{dy}{dx} = x^2 - 5x + 7 = 1$</span></p><p><strong>Step 4:</strong> Solve: <span class='math'>$x^2 - 5x + 6 = 0$</span> gives <span class='math'>$x = 2$</span> or <span class='math'>$x = 3$</span></p><p><strong>Step 5:</strong> Find corresponding y-values by substituting in f(x). ∴ Answers are (a) and (b).</p>
Correct Answer: A, B