Question

Define the function h:RR by h(x)=x44x3+3x22.

Show that there is some cR such that h(x)h(c)xR.

You may use the Extreme Value Theorem (EVT).

Hint: You apply EVT to h (which is a polynomial, thus continuous), over a closed interval and then obtain some cR s.t. h(x)h(c)cR

Below is a similar question, which is partly answered below. The above question should be answered in a similar way.

Define the function h:RR by h(x)=x63x3+2x21,

Show that there is some cR such that h(x)h(c)xR.

Part Answer (please complete the rest as well as the above question)

Apply the EVT to h (which is a polynomial and therefore continuous) over the closed interval [2,2]. Why is the interval [2,2] appropriate ?

Why is the following useful ?

x>2x3>8x3x3>8x3x6>8x3(sincetheexponent6isevenonecandropthemodulus)

and then for the rest of the polynomial, why is this useful ?

3x3+2x213x3+2x2+13x3+2x213x3+2x2+13x3+2x3+x3=6x3

*Note that h(0)=1. For x>2, we have x6>8x3, while

3x3+2x216x3.

So h(x)>0>1>=h(0)h(c). Hence h(x)h(c)xR

Please explain *Note