Science:Math Exam Resources/Courses/MATH110/December 2013/Question 06 (c)

MATH110 December 2013

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Question 06 (c)
Consider the function: $f(x)={\dfrac {e^{\left(x^{3}\right)}e^{\left(2x^{2}\right)}e^{x}}{e^{3}}}.$ (c) How many solutions are there to the equation $\displaystyle f'(x)=0$? Justify your answer.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The derivative was computed in part (b): $\displaystyle f'(x)=\left(3x^{2}+4x+1\right)\cdot e^{x^{3}+2x^{2}+x-3}.$ Now set the derivative and solve. This can be done either by factoring or using the quadratic formula.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Recall that $\displaystyle f'(x)=\left(3x^{2}+4x+1\right)\cdot e^{x^{3}+2x^{2}+x-3}.$ From part (a), we know that $e^{x^{3}+2x^{2}+x-3}\neq 0$. That implies that $\displaystyle f'(x)=0$ when $3x^{2}+4x+1=0$. Factoring, we get $(3x+1)(x+1)=0$, which means $x=-{\frac {1}{3}},-1$. That means there are two solutions for when $\displaystyle f'(x)=0$.