Stationary Points

f($x$) = $x^2-4x+3$. Find and classify its stationary point.

• 1Differentiate: f'($x$) = $2x-4$
• 2Set f'($x$) = $0$: $2x-4=0$, so $x=2$
• 3Check the sign either side: f'($1$) = $2$($1$) − $4=-2$ (negative), f'($3$) = $2$($3$) − $4=2$ (positive)
• 4Negative then positive: this is a minimum
• 5Find the $y$-value: f($2$) = $4-8+3=-1$

Answer: Minimum point at ($2$, $-1$)

f($x$) = $-x^2+4x$. Find and classify its stationary point.

• 1Differentiate: f'($x$) = $-2x+4$
• 2Set f'($x$) = $0$: $-2x+4=0$, so $x=2$
• 3Check the sign either side: f'($1$) = $-2$($1$) + $4=2$ (positive), f'($3$) = $-2$($3$) + $4=-2$ (negative)
• 4Positive then negative: this is a maximum
• 5Find the $y$-value: f($2$) = $-4+8=4$

Answer: Maximum point at ($2$, $4$)