What is the equation of the normal line

Elijah Schwartz

Elijah Schwartz

Answered question

2022-04-09

What is the equation of the normal line of f(x)=(x3)3(x+2) at x=0?

Answer & Explanation

xcentricccrhb1

xcentricccrhb1

Beginner2022-04-10Added 12 answers

Expanding the formula for f gives:
f(x)=(x39x2+27x27)(x+2)
=x49x3+27x227x+2x318x2+54x54
=x47x3+9x2+27x54.
Therefore, f(x)=4x3212+18x+27.
Also notethat f(0)=-54 and f'(0)=27.
The normal line at the point (x,y)=(0, f(0))=(0, -54) has slope equal to the negative reciprocal of f'(0)=27, so this slope equals 127.
Since this normal line goes through the point (0,-54), it follows that its equation is y=127x54

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