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dalvyx [7]
3 years ago
8

Diana worked on her science project for 6 1 3 hours. Gabe worked on his science project 1 3 4 times as long as Diana. Paula work

ed on her science project 3 4 times as long as Diana. Select True or False for each statement. Diana worked longer on her science project than Gabe worked on his science project.
Mathematics
1 answer:
Vaselesa [24]3 years ago
4 0
Are your numbers right?  Diana worked on her project for 613 hours, Gabe worked on it for 134 times as long as Diana, and Paula worked on her science project 34 times as long as Diana?  Assuming they are, It is false that Diana worked longer on her science project than Gabe worked on his.  Tell me if I read the question wrong, I am happy to help.
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Part A:

Given the function f(x)=x+2\cos(x), the absolute maximum or minimum occurs when f'(x)=0.

f'(x)=0 \\  \\ \Rightarrow1-2\sin{x}=0 \\  \\ \Rightarrow2\sin{x}=1 \\  \\ \Rightarrow\sin{x}= \frac{1}{2}  \\  \\ \Rightarrow x=\sin^{-1}{\frac{1}{2}}= \frac{\pi}{6}

Using the second derivative test,

f''(x)=-2cosx \\  \\ \Rightarrow f''\left( \frac{\pi}{6} \right)=-2\cos{\left( \frac{\pi}{6} \right)}=-1.732

Since the second derivative gives a negative number, the given function has a maximum point at x=\frac{\pi}{6}.

And the maximum point is given by:

f\left( \frac{\pi}{6} \right)=\frac{\pi}{6}+2\cos\left( \frac{\pi}{6} \right) \\  \\ =0.5236+2(0.8660)=0.5236+1.732 \\  \\ =\bold{2.256}

i.e. \left(\frac{\pi}{6},\ 2.256\right)



Part B:

Given the function f(x)=e^{-x}-e^{-2x}, the absolute maximum or minimum occurs when f'(x)=0.

f'(x)=0 \\ \\ \Rightarrow-e^{-x}+2e^{-2x}=0 \\ \\ \Rightarrow2e^{-2x}=e^{-x} \\ \\ \Rightarrow2e^{-x}=1 \\ \\ \Rightarrow e^{-x}=\frac{1}{2} \\  \\ \Rightarrow-x=\ln \frac{1}{2}=-0.6931 \\  \\ \Rightarrow x=0.6931

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f''(x)=e^{-x}-4e^{-2x} \\ \\ \Rightarrow f''(0.6931)=e^{-0.6931}-4e^{-2(0.6931)} \\  \\ =0.5-4e^{-1.386}=0.5-4(0.25)=0.5-1 \\  \\ =-0.5

Since the second derivative gives a negative number, the given function has a maximum point at x=0.6931.

And the maximum point is given by:

f(0.6931)=e^{-0.6931}-e^{-2(0.6931)} \\  \\ =0.5-e^{-1.386}=0.5-0.25=\bold{0.25}

i.e. (0.693, 0.25)
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