How to solve the critical point of a function Local minimum local maximum

Question

ScanSolve AI Tutor · Accepted Answer

To find the critical points, local minima, and local maxima of a function $f(x)$ :

Find the first derivative of the function, $f'(x)$ .
$f'(x) = \frac{d}{dx} f(x)$
Find the critical points by setting the first derivative equal to zero and solving for $x$ , or by finding where the first derivative is undefined.
$f'(x) = 0 \quad \text{or} \quad f'(x) \text{ is undefined}$
The $x$ -values obtained are the critical points.
Use either the First Derivative Test or the Second Derivative Test to classify these critical points as local minima, local maxima, or neither.
- First Derivative Test:
  
  Choose test values in intervals around each critical point.
  
  If $f'(x)$ changes from positive to negative at a critical point, it's a local maximum.
  
  If $f'(x)$ changes from negative to positive at a critical point, it's a local minimum.
  
  If $f'(x)$ does not change sign, it's neither (often an inflection point).
- Second Derivative Test:
  
  Find the second derivative of the function, $f''(x)$ .
  
  Substitute each critical point $x_c$ into $f''(x)$ .
  
  If $f''(x_c) > 0$ , the critical point is a local minimum.
  
  If $f''(x_c) < 0$ , the critical point is a local maximum.
  
  If $f''(x_c) = 0$ , the test is inconclusive, and you should use the First Derivative Test.

Step-by-step solution