Question on intersection of tangent and chord

[Titan97]

Homework Statement

Show that The tangent at (c,e^c) on the curve y=e^x intersects the chord joining the points (c-1,e^c-1) and (c+1,e^c+1) at the left of x=c

Homework Equations

Legrange's mean value theorem

The Attempt at a Solution

f'(c)=e^c
Applying LMVT at c-1, c+1
$$f'(a)=\frac{e^c(e-\frac{1}{e})}{2}\ge f'(c)$$
Hence the chord is parallel to the tangent at $a$ and for e^x, if f'(a)>f'(c) then a>c.
So chord has a slope greater than the slope of tangent at c. Hence it intersects at left of x=c.
Is this correct? Are there any other methods?

 

[Mark44]

Homework Statement

Show that The tangent at (c,e^c) on the curve y=e^x intersects the chord joining the points (c-1,e^c-1) and (c+1,e^c+1) at the left of x=c

Homework Equations

Legrange's mean value theorem

That's Lagrange.

The Attempt at a Solution

f'(c)=e^c
Applying LMVT at c-1, c+1
$$f'(a)=\frac{e^c(e-\frac{1}{e})}{2}\ge f'(c)$$

What is a? You haven't said what it is.

Hence the chord is parallel to the tangent at $a$ and for e^x, if f'(a)>f'(c) then a>c.
So chord has a slope greater than the slope of tangent at c. Hence it intersects at left of x=c.
Is this correct? Are there any other methods?

You could find the equation of the tangent line at (c, e^c) and find the equation of the chord through the two other points, and show that the intersection of the tangent line and chord are at a value of x less than c. That's the approach I would take, but I haven't gone all the way through to see if it is fruitful.

 

[BvU]

I think this is what the composer of the exercise meant you to do. Don't see any other inroad.

 

[Titan97]

What is a? You haven't said what it is.

Some x value between c-1 and c+1which satisfies mvt.