y=e^2x-11e^x+24 Find the stationary point, nature of the stationary point, the x-intercepts and the y-intercept (calculator allowed)
Stationary point:
dy/dx = 2e^2x - 11e^x =0
2e^2x = 11e^x
e^x=5.5 (can divide by e^x since e^x > 0 for all x)
x=ln(5.5), y=5.5^2-115.5+24=-6.25
Answer: (ln(5.5),-6.25)
Nature of stationary point:
Evaluate y'' at x=ln(5.5)
y''= 4e^2x-11e^x = 45.5^2-11*5.5 = 60.5
Since y''>0, local minima
x-intercept:
Set y=0
y=(e^x-3)(e^x-8)=0
e^x=3 or e^x=8
x=ln(3) and x=ln(8)=3ln(2)
Answers: (ln(3),0) and (3ln(2),0)
y-intercept:
Set x=0
y=1-11+24=14
y-intercept: (0,14)
AJ
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