Step-by-Step Solution
Step 1: Analyze the Leading Coefficient
Comparing with f(x)=ax2+bx+c:
a=−1,b=−3,c=−5
Because a=−1<0, the parabola opens downwards (∩). Therefore, the function achieves an absolute maximum value at its vertex.
Step 2: Complete the Square
Factor out −1 from the x-terms:
f(x)=−(x2+3x)−5
Complete the square inside the parenthesis by adding (23)2=49. Balancing outside by adding 49:
f(x)=−(x2+3x+49)−5+49=−(x+23)2−420+49=−(x+23)2−411
Step 3: Determine the Extremum Point
Because −(x+23)2≤0 for all real x, the maximum is achieved when the squared expression vanishes:
x+23=0⟹x=−23=−1.5
Step 4: Calculate the Maximum Value
Substituting x=−23:
f(−23)=−0−411=−411=−2.75
