Show that the curve y=2ex+3x+5x3 has no tangent line with slope 2.

Question

sukljama2 · Accepted Answer

y=2ex+3x+5x3y′=d[2ex+3x+5x3]dxy′=2ex+3⋅1+5⋅3x3−1Therefore, slope of the tangent at x=a is given by m(a)=2ea+3+15a2The slope of the tangent will be 2 when2ea+3+15a2=2Subtract 3 from both sides2ea+15a2=−1Note that:(1) 2ea is always non-negative(2) 15a2 is always non-negative(3) Sum of two non-negative numbers cannot be negative.Therefore, 2ea+15a2=−1 has no solutions and no such tangent exists.Hint: slope of the tangent at x=a is given by m(a)=2ea+3+15a2

Bertha Jordan · Accepted Answer

y′=2ex+3+5(3x2)=2ex+3+15x2Verify that y&#039; cannot be equal to 2 which shows that no tangent line can have slope 2.If&amp;nbsp;y′=2⇒2ex+3+15x2=2⇒2ex(positive)+1+15x2(non−¬ative)=0&amp;nbsp;(*)The expression&amp;nbsp;2ex+1+15x2&amp;nbsp;- positive, so it cannobe equal to zero, so the equality in (*) is impossible, therefore there is no point on the curve where the slope is 2.

user_27qwe · Accepted Answer

y=2ex+3x+5x31) take the derivative of your function with respect to x . .y′=2ex+3+15x22) derivative means slope of a tangent line so we can see that derivative (slop of tangent line) more than 2 .y′&amp;gt;2how do we know it is larger than to ?2ex&amp;gt;03&amp;gt;2 and 15x2&amp;gt;or=0 so no mater what x value we plug in our y&#039; value will be more than 2.

Show that the curve y = 2e^{x} + 3x +

Answered question

Answer & Explanation

New Questions in Precalculus