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Chapter 3: Q. 93 (page 277)

Prove that every quadratic function is either always concave up or always concave down.

Short Answer

Expert verified

Proved that every quadratic function is either always concave up or always concave down.

Step by step solution

01

Step 1. Given Information.

Given a quadratic function. Let that function bef(x)=ax2+bx+c.

02

Step 2. Theorem.

The Derivative Measures Where a Function is Increasing or Decreasing

Let f be a function that is differentiable on an interval I.

(a) If f'is positive in the interior of I, then f is increasing on I.

(b) If f'is negative in the interior of I, then f is decreasing on I.

(c) If f'is zero in the interior of I, then f is constant on I.

03

Step 3. Proof.

f(x)=ax2+bx+c,Differentiatingbothsidesw.r.txweget,f'(x)=2ax+b,Differentiatingbothsidesw.r.txonceagainweget,f''(x)=2a,Now,f''(x)isaconstantanddoesnotdependonxandaccordingtothevalueofconstantaitwillbeeitherpositiveornegativeandaccordinglythefunctionf(x)willbeconcaveupwardorconcavedownwardrespectively.

Hence, Proved.

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Most popular questions from this chapter

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=x2+3x

Find the critical points of the function

f'(x)=(x-1)(x-2)x-3

Determine the graph of a function f from the graph of its derivative f'.

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=xx2+1

Use the definition of the derivative to find f' for each function f.

f(x)=x3+2
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