Cubic with Adjustable Coefficients
Application of Derivatives · Class XII
Shape a cubic y = ax³ + bx² + cx + d. Find critical points and use the second derivative test.
- Discriminant of f′ = 3ax² + 2bx + c36
- Critical x₁-1
- Critical x₂1
- y = a x³ + b x² + c x + d
- y′ = 3ax² + 2bx + c
Formulas in this lab
- Discriminant of f′ = 3ax² + 2bx + c
- Critical x₁
- Critical x₂
Frequently asked questions
▶What is the second derivative test?
At a critical point $x_0$ where $f'(x_0) = 0$: if $f''(x_0) > 0$, it's a local minimum; if $f''(x_0) < 0$, a local maximum; if $f''(x_0) = 0$, the test is inconclusive. So for $f(x) = x^3 - 3x$, $f'(\pm 1) = 0$, $f''(1) = 6 > 0$ (min), $f''(-1) = -6 < 0$ (max).
▶How do I use the cubic coefficients lab?
Slide $a, b, c, d$ in $f(x) = ax^3 + bx^2 + cx + d$. The lab plots $f$ and $f'$ together and reports the discriminant of $f'$ and the critical points. Where the red $f'$ crosses zero, blue $f$ has an extremum.
▶When does the second derivative test fail?
When $f''(x_0) = 0$, e.g. for $f(x) = x^4$ at $x_0 = 0$. Use the first derivative test or higher derivatives. Students apply $f''$ blindly and miss inflection points. The lab lets you sweep to $a = 0$ (degenerate cubic) and feel the breakdown.
▶Where is the second derivative test used in real life?
Profit maximization in economics requires marginal cost = marginal revenue ($f' = 0$) plus a max condition ($f'' < 0$). Engineering finds stress concentration peaks the same way. The cubic in this lab is a sandbox for these optimization patterns.