Show that the curve y = 2*e^x +3x + 5x^3 has no tangent line with slope 2.
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bookM.S. from University of Kerala , India
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We have to show that the curve `y=2e^x+3x+5x^3` has no tangent line with slope 2.
Now, we know that the slope of the tangent is found by taking the first derivative of the curve. In other words,
`\frac{dy}{dx}=2e^x+3+15x^2`
Now, let us consider that there is a tangent line whose slope is 2.
To demonstrate,
`\frac{dy}{dx}=2`
`2e^x+3+15x^2=2`
For example, `2e^x+15x^2=-1`
But we know that `e^x ` and `x^2` are always positive. So the left hand side of the above equation will be always positive, but the right hand side is always negative.
Hence, there are no values of x for which we get the slope 2.
Therefore we can say that there are no tangent lines to the given curve with a slope of 2.
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We are asked to show that curve represented by the function `y=2e^x+3x+5x^3 ` has no tangent line with a slope of 2:
The slope of a tangent line to a curve at a point is found by evaluating the first derivative of the function at that point.
Here the first derivative is:
`y'=2e^x+3+15x^2 `
If we temporarily assume that...
(The entire section contains 4 answers and 604 words.)
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