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Chapter 10: Problem 794

If \(\mathrm{f}(\mathrm{x})=\mathrm{Ax}+\mathrm{B}\) and \(\mathrm{f}(0)=\mathrm{f}^{\prime}(\mathrm{x})=2\) then \(\mathrm{f}(1)=\) (a) 4 (b) 2 (c) 1 (d) \(-4\)

Short Answer

Expert verified
The correct answer is \(f(1) = 4\).

Step by step solution

01

Differentiate the function

In order to find the derivative of the function \(f(x) = Ax + B\), we will differentiate with respect to x. The derivative of \(Ax\) with respect to x is A, and the derivative of a constant (B) is 0. Therefore, the derivative of \(f(x)\) is given by: \[f'(x) = A\]
02

Use the given conditions to solve for A and B

We know that \(f(0) = 2\) and \(f'(x) = 2\). Using these conditions, we can solve for A and B: First condition: \(f(0) = 2\) Plug in the x-value (0) into the function: \(f(0) = A(0) + B = 2\) \(B = 2\) Second condition: \(f'(x) = 2\) Since the derivative is a constant value of A, we can equate it to the given condition: \(f'(x) = A = 2\) So, A = 2 and B = 2.
03

Find the value of f(1)

Now that we have found the values of A and B, we can find the value of \(f(1)\) using the linear function: \(f(x) = Ax + B\) \(f(1) = 2(1) + 2 = 4\) Therefore, \(f(1) = 4\). The correct answer is (a) 4.

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Most popular questions from this chapter

If \(\mathrm{y}=\tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+\mathrm{x}+1\right)\right] \tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+3 \mathrm{x}+3\right)\right]\) \(+\tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+5 \mathrm{x}+7\right)\right] \ldots \ldots\) to \(\mathrm{n}\) terms then \((\mathrm{dy} / \mathrm{d} \mathrm{x})\) \(=\) (a) \(\left[1 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]+\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (b) \(\left[1 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]-\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (c) \(\left[2 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]+\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (d) \(\sum \mathrm{n}\)

The Roll's theorem is applicable in the interval \(-1 \leq \mathrm{x} \leq 1\) for the function (a) \(\mathrm{f}(\mathrm{x})=\mathrm{x}\) (b) \(f(x)=x^{2}\) (c) \(f(x)=2 x^{2}+3\) (d) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}|\)

If \(\mathrm{y}=\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))\) and \(\mathrm{f}(0)=0, \mathrm{f}^{\prime}(0)=1\) then \(\mid[\mathrm{dy} / \mathrm{d} \mathrm{x}]_{\mathrm{x} 0}=\) (a) 0 (b) 1 (c) - 1 (d) 2

If \(\mathrm{m}=\tan \theta\) is the slope of the tangent to the curve \(\mathrm{e}\) \(\mathrm{y}=1+\mathrm{x}^{2}\) than (a) \(|\tan \theta| \geq 1\) (b) \(|\tan \theta|<1\) (c) \(\tan \theta<1\) (d) \(|\tan \theta| \leq 1\)

If \(f(x)=x \cdot \cot ^{-1} x\) then \(f^{\prime}(1)=\) (a) \((\pi / 4)-(1 / 2)\) (b) \((\pi / \overline{4)+(1 / 2)}\) (c) \((\pi / 4)-(1 / 3)\) (d) \((\pi / 4)-1\)
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