Find the minimum and maximum values of the function $f(x)=x^2-4x+5$ on the interval $[1,4]$.

Step-by-step solution

1. Compute the derivative: $f^{\prime }(x)=2x-4$; this is zero when $2x-4=0$, so $x=2$.
2. Evaluate $f$ at the critical point and at the endpoints: $f(1)=2$, $f(2)=1$, $f(4)=5$.
3. The smallest value is $1$ at $x=2$ and the largest is $5$ at $x=4$.
4. Therefore the function attains a minimum of $1$ at $x=2$ and a maximum of $5$ at $x=4$.
5. The answer is the pair: maximum $5$ at $x=4$ and minimum $1$ at $x=2$.

Concept tested: Find extrema of a quadratic on a closed interval

Published 2 September 2026 · How this solution was prepared and checked

Related questions

• 1 A rectangle is to be inscribed in a right‑angled triangle whose base is 12,cm and height is 8,cm. The rectangle’s base lies on the triangle’s base and its top side is parallel to that base. Find the dimensions of the rectangle that give the maximum area.
• 2 A company has the cost function C(x)=1000+50x+2x^2 rupees, where x is the number of units produced in thousands. Determine the production level x that minimises the average cost and compute the minimum average cost.
• 3 Water is being poured into a conical tank of height 12 m and radius 4 m. The water level rises at a constant rate of 0.5 m/min. Find the rate of increase of the volume of water when the water level is 6 m.
• 4 Evaluate the definite integral ∫_0^(1/2) dx/√(1-4x^2).
• 5 Find the domain of the function f(x)=tan^-1(√(1-x^2)).
• 6 Find the magnitude of the vector u=3i-4j+12k.