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Chapter 3: Q. 88 (page 263)

Prove that every quadratic function has exactly one local extremum.

Short Answer

Expert verified

It is proded that every quadratic function has exactly one local extremum.

Step by step solution

01

Step 1. Given Information 

We are given that every quadratic function has exactly one local extremum.

02

Step 2. Proving the statement 

Consider the quadratic function,

f(x)=ax2+bx+c;wherea鈮�0.

The objective is to prove that f has exactly one local extremum. The first derivative of the function is given by,

f'(x)=ddxax2+bx+c=2ax+b

For critical values,

role="math" localid="1648437135068" f'(x)=02ax+b=02ax=-bx=-b2a

As x has only one real value, The function has exactly one local extremum. Therefore, every quadratic function has exactly one local extremum.

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Most popular questions from this chapter

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

Restate the Mean Value Theorem so that its conclusion has to do with tangent lines.

Determine whether or not each function f in Exercises 53鈥�60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=x2鈭�1x2鈭�4,[a,b]=[鈭�3,3]

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=ex(x-2)
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