GATE CS 2008 Q25 — Maxima and Minima, Mean Value Theorem, Integration

To find the number of distinct extrema for the curve $f(x) = 3x^4 - 16x^3 + 24x^2 + 37$, we first find its critical points by setting the first derivative to zero.

Differentiating $f(x)$ with respect to $x$:

Factorizing the derivative:

Setting $f'(x) = 0$ gives the critical points:

Now, let's analyze the nature of these critical points using the first derivative test:

• At $x = 0$:
  • For $x < 0$, $f'(x) < 0$ (since $x$ is negative and $(x-2)^2$ is positive).
  • For $0 < x < 2$, $f'(x) > 0$ (since $x$ is positive and $(x-2)^2$ is positive).
  Since $f'(x)$ changes sign from negative to positive as $x$ increases through $0$, $x = 0$ is a local minimum (an extremum).


• At $x = 2$:
  • For $x$ slightly less than $2$ (e.g., $1.9$), $f'(x) > 0$.
  • For $x$ slightly greater than $2$ (e.g., $2.1$), $f'(x) > 0$.
  Since $f'(x)$ does not change sign as $x$ passes through $2$, $x = 2$ is a point of inflection, not an extremum.

Thus, there is only $1$ distinct extremum (at $x = 0$).