Find the maximum value of $3-x^2+2x+3$.See answer

Daftar Isi

Find the maximum value of 3 x2 2x 3

Question

Find the maximum value of $3-x^2+2x+3$.

Super Answer

Let me help you find the maximum value of this quadratic function.

Step 1: Identify the Quadratic Function

• Function: $f(x)=3-x^2+2x+3$
• Rearrange to standard form: $f(x)=-x^2+2x+6$
• Leading coefficient $a=-1$ (opens downward, has maximum)

Step 2: Find Critical Point Using Derivative

• $f^{\prime }(x)=-2x+2$
• Set $f^{\prime }(x)=0$: $-2x+2=0$
• Solve for x: $x=1$

Step 3: Verify Maximum

• Since $a=-1<0$, the critical point is a maximum
• The vertex occurs at $x=1$

Step 4: Calculate Maximum Value

• Substitute $x=1$ into original function
• $f(1)=3-(1{)}^2+2(1)+3$
• $f(1)=3-1+2+3$
• $f(1)=7$

Final Answer

The maximum value of $3-x^2+2x+3$ is 7, occurring at $x=1$.