Consider the function $f(x)=x^2-4x+3$. Find its minimum value on the interval $[1,3]$.

Step-by-step solution

1. 1. Compute the first derivative: $f^{\prime }(x)=2x-4$.
2. 2. Set the derivative to zero: $2x-4=0\to x=2$, which lies within $[1,3]$.
3. 3. Evaluate the function at the critical point: $f(2)=2^2-4\cdot 2+3=-1$.
4. 4. Check the endpoints: $f(1)=0$ and $f(3)=0$.
5. 5. The smallest value among $-1,0,0$ is $-1$.

Concept tested: Minimum of a quadratic using derivative

Published 4 October 2026 · How this solution was prepared and checked