# Which of the following is larger?

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• Oct 21st 2009, 03:38 PM
Latszer
Which of the following is larger?
Suppose f has a continuous positive second derivative for all x. Which of the following is larger?
f(1 + deltax)
f(1) + f '(1)*deltax

I need an explanation, my first instinct is to think that it is the first choice because of the positive second derivative.
• Oct 21st 2009, 04:54 PM
skeeter
Quote:

Originally Posted by Latszer
Suppose f has a continuous positive second derivative for all x. Which of the following is larger?
f(1 + deltax)
f(1) + f '(1)*deltax

I need an explanation, my first instinct is to think that it is the first choice because of the positive second derivative.

\$\displaystyle f(1) + f'(1)(\Delta x)\$ is a tangent line approximation of \$\displaystyle f(1 + \Delta x)\$

\$\displaystyle f(x)\$ is concave up ... a tangent line approximation of \$\displaystyle f(1 + \Delta x)\$ would be less than the actual value of \$\displaystyle f(1 + \Delta x)\$ because the tangent line lies beneath the curve.