Critical points

Refer to PennCalc Main/Optimization

For analyzing a function, it’s very efficient to have a look at its Critical points, which could be classified as Extrema, Inflection, Corner, and Discontinuity.

How to find critical points

Strategy:

• Knowing that f(x) has critical point c when f'(c) = 0 or f'(c) is undefined
• Differentiate f(x) to get f'(x)
• Solve c for f'(c)=0 & f'(c) undefined

Refer to Symbolab’s step-by-step solution.%3Dx%5Ccdot%20sqrt%5Cleft(4-x%5Cright))

Example

Solve:

• See that original function f(x) is undefined at x = 2 or -2
• Differentiate f(x) to get f'(x):

• Solve f'(x)=0 only when x=0.
• f'(x) is undefined when x=2 or -2, as the same with f(x) so it's not a solution.

Example

Solve: Refer to Symbolab step-by-step solution.%3Dx%5Ccdot%20sqrt%5Cleft(4-x%5Cright))

• Differentiate f(x) to get f'(x):

• f'(x) is undefined when x > 4
• Solve f'(x)=0 get x = 8/3
• So under the given condition, only x=8/3 is the answer.