15. -/1 points SCalcET8 4.3.015. Consider the equation below. (If an answer does

Question

15. -/1 points SCalcET8 4.3.015. Consider the equation below. (If an answer does not exist, enter DNE.) (a) Find the interval on which fis increasing. (Enter your answer using interval notation.) Find the interval on which fis decreasing. (Enter your answer using interval notation.) (b) Find the local minimum and maximum values of f. local minimum value local maximum value (c) Find the inflection point. (x, y) = Find the interval on which fis concave up. (Enter your answer using interval notation.) Find the interval on which fis concave down. (Enter your answer using interval notation.) Need Help?

Explanation / Answer

f(x) = e^(7x) + e^(-x)

DEriving :
f'(x) = 7e^(7x) - e^(-x) = 0
7e^(7x) = e^(-x)
e^(8x) = 1/7
8x = -ln(7)
x = -ln(7) / 8

Region 1 : (-inf , -ln7/8) :
Test = -2
f'comes out -7.3890502782296165, i.e negative
So, decreasing

Region 2 : (-ln7/8) , inf)
Test = 1
f'(1) = 7e^7 - e^-2 , positive surely
So, inc

Inc : (-ln(7)/8 , inf)
Dec : (-inf , -ln(7)/8)

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Clearly x = -ln(7)/8 is a pt of minimum
cus dec changes to inc there

When x = -ln(7)/8, we get
1.4576

So, local min val = 1.4576
local max val = DNE

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c) Inflection :
f'(x) = 7e^(7x) - e^(-x)

Deriving again :
f''(x) = 40e^(7x) + e^(-x) = 0
No solution

So, no inf pts

DNE

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Concvity :
Clearly f''(x) = 40e^(7x) + e^(-x) is positive everywhere
because both exponential terms are positive and gettin added together

So, concave up : (-inf , inf)
conc down : DNE

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