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Solve the given problems.The second derivative of a function is $12 x^{2}$. Explain how to find the function if its curve passes through the points (1,6) and (2,21) Find the function.

Calculus 1 / AB

Chapter 25

Integration

Section 2

The Indefinite Integral

Integrals

Missouri State University

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University of Nottingham

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So here we're as to find the derivative of the function. Why equals negative to X rays to the one point five plus twelve times X rays to the point five. Now we know that the derivative off some or difference of functions the same as summer difference off the derivatives. So in this case, we get d Y over the X equals the over DX off negative two X rays to the one point five plus the over DX off twelve x rays to the point five. Now, in this case, in both of these derivatives, negative two and twelve are constants. We know that the derivative off a constant times a function. It's same as the constant times. Terrible live off the function, meaning we can just pull out the Constance out of the derivative. So we get negative two times the over DX off X rays to the one point five plus twelve times the over DX off X rays to the point five. Now, to find these two derivatives, we are going to use the power rule. The power rule states that derivative off X rays to the end, equals end times X rays to the n minus one here and can take any values except negative one. So in these Stradivari lives, the value's off our one point five and point five. So we are going to use the power rule to find these derivatives so negative two stays as it is now derivative off X rays to the one point five and is one point five. So that becomes one point five X rays to the one born five minus one plus twelve stays as it is derivative off X rays to the point five here and this point five. So we get zero point five times X rays to the zero point five minus one, so we can simplify this further negative. Two times one point five becomes negative. Three X rays to the one point five minus one is just point five plus twelve times point five is six X rays to the point five minus one is X rays to the negative point five. So my derivative off the given function is D Y. Over DX equals negative three times X rays to the point five plus six times X rays to the negative point five

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