4748 One way of proving that f(x) g(x) for all x in a giveninterval is to show that 0

Chapter 4, Problem 48

(choose chapter or problem)

One way of proving that \(f(x) \leq g(x)\) for all \(x\) in a given interval is to show that \(0 \leq g(x)-f(x)\) for all \(x\) in the interval; and one way of proving the latter inequality is to show that the absolute minimum value of \((x)-f(x)\) on the interval is nonnegative. Use this idea to prove the inequalities in these exercises.

                        

Prove that \(\cos x \geq 1-\left(x^{2} / 2\right)\) for all \(x\) in the interval \([0,2 \pi]\).

Equation Transcription:

Text Transcription:

f(x) leq g(x)

x

0 leq g(x)-f(x)

x

g(x)-f(x)

cos x geq 1-(x^2/2)

x

[0,2pi]