Optimization from a Given Model - Calculus Step by Step

Minimize C(x)=x²-6x+11 by finding where C'(x)=2x-6=0, giving x=3, C(3)=2.

optimization-model-given To optimize a given function: find critical points by setting C'(x)=0, then use C''(x) to classify (positive means minimum, negative means maximum), and evaluate C at the critical point to find the optimal value.

Example

Find and classify the critical point of a given objective model.

highlighted = computed this step

Step 1 — Set up model

Start with the objective model.

C(x)= x^{2}-6x+11

Step 2 — Differentiate

Differentiate the objective.

C'(x)= \hlmath{2x-6}

Step 3 — Critical point

Set C prime equal to 0 to get x equals 3.

2x-6 = 0 \Rightarrow x= \hl{3}

Step 4 — Classify

Because C double prime is positive, this is a minimum.

C''(x)= \hlmath{2} > 0 \Rightarrow \hl{minimum}

Step 5 — Optimum value

Evaluate C at 3 to get optimum value 2.

C( 3 )= \hl{2}