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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

Restate Theorem 3.3 so that its conclusion has to do with

tangent lines.

It took Alina half an hour to drive to the grocery store that is 20 miles from her house.

(a) Use the Mean Value Theorem to show that, at some point during her trip, Alina must have been traveling exactly 40 miles per hour.

(b) Why does what you have shown in part (a) make sense in real-world terms?

Determine whether or not each function fsatisfies 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∈(a,b)that satisfy the conclusion of the Mean Value Theorem.

f(x)=tanx,[a,b]=[-π,π].

Determine whether or not each function f in Exercises 53–60 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 ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=−x3+3x2−7,[a,b]=[−2,3]

For the graph of f in the given figure, approximate all the values x ∈ (0, 4) for which the derivative of f is zero or does not exist. Indicate whether f has a local maximum, minimum, or neither at each of these critical points.
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